X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_algebra.py;h=2c689d79d3c85d5d6f4d0daca39ca3f0e649540d;hb=fc29add6cf1d9ff4e8a240b0f8f2ca6672d4ea57;hp=ec58b57e8d78b004bcdc0ffaa83fc773c3cfffa8;hpb=ee3262f5130f2e7b38b520d4238cede0cea9b981;p=sage.d.git diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py index ec58b57..106a0cd 100644 --- a/mjo/eja/eja_algebra.py +++ b/mjo/eja/eja_algebra.py @@ -1,254 +1,725 @@ """ -Euclidean Jordan Algebras. These are formally-real Jordan Algebras; -specifically those where u^2 + v^2 = 0 implies that u = v = 0. They -are used in optimization, and have some additional nice methods beyond -what can be supported in a general Jordan Algebra. +Representations and constructions for Euclidean Jordan algebras. + +A Euclidean Jordan algebra is a Jordan algebra that has some +additional properties: + + 1. It is finite-dimensional. + 2. Its scalar field is the real numbers. + 3a. An inner product is defined on it, and... + 3b. That inner product is compatible with the Jordan product + in the sense that ` = ` for all elements + `x,y,z` in the algebra. + +Every Euclidean Jordan algebra is formally-real: for any two elements +`x` and `y` in the algebra, `x^{2} + y^{2} = 0` implies that `x = y = +0`. Conversely, every finite-dimensional formally-real Jordan algebra +can be made into a Euclidean Jordan algebra with an appropriate choice +of inner-product. + +Formally-real Jordan algebras were originally studied as a framework +for quantum mechanics. Today, Euclidean Jordan algebras are crucial in +symmetric cone optimization, since every symmetric cone arises as the +cone of squares in some Euclidean Jordan algebra. + +It is known that every Euclidean Jordan algebra decomposes into an +orthogonal direct sum (essentially, a Cartesian product) of simple +algebras, and that moreover, up to Jordan-algebra isomorphism, there +are only five families of simple algebras. We provide constructions +for these simple algebras: + + * :class:`BilinearFormEJA` + * :class:`RealSymmetricEJA` + * :class:`ComplexHermitianEJA` + * :class:`QuaternionHermitianEJA` + * :class:`OctonionHermitianEJA` + +In addition to these, we provide two other example constructions, + + * :class:`JordanSpinEJA` + * :class:`HadamardEJA` + * :class:`AlbertEJA` + * :class:`TrivialEJA` + +The Jordan spin algebra is a bilinear form algebra where the bilinear +form is the identity. The Hadamard EJA is simply a Cartesian product +of one-dimensional spin algebras. The Albert EJA is simply a special +case of the :class:`OctonionHermitianEJA` where the matrices are +three-by-three and the resulting space has dimension 27. And +last/least, the trivial EJA is exactly what you think it is; it could +also be obtained by constructing a dimension-zero instance of any of +the other algebras. Cartesian products of these are also supported +using the usual ``cartesian_product()`` function; as a result, we +support (up to isomorphism) all Euclidean Jordan algebras. + +SETUP:: + + sage: from mjo.eja.eja_algebra import random_eja + +EXAMPLES:: + + sage: random_eja() + Euclidean Jordan algebra of dimension... """ - - -from sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra import FiniteDimensionalAlgebra -from sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra_element import FiniteDimensionalAlgebraElement from sage.algebras.quatalg.quaternion_algebra import QuaternionAlgebra -from sage.categories.finite_dimensional_algebras_with_basis import FiniteDimensionalAlgebrasWithBasis -from sage.functions.other import sqrt +from sage.categories.magmatic_algebras import MagmaticAlgebras +from sage.categories.sets_cat import cartesian_product +from sage.combinat.free_module import CombinatorialFreeModule from sage.matrix.constructor import matrix +from sage.matrix.matrix_space import MatrixSpace from sage.misc.cachefunc import cached_method -from sage.misc.prandom import choice -from sage.modules.free_module import VectorSpace -from sage.modules.free_module_element import vector -from sage.rings.integer_ring import ZZ -from sage.rings.number_field.number_field import QuadraticField -from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing -from sage.rings.rational_field import QQ -from sage.structure.element import is_Matrix -from sage.structure.category_object import normalize_names +from sage.misc.table import table +from sage.modules.free_module import FreeModule, VectorSpace +from sage.rings.all import (ZZ, QQ, AA, QQbar, RR, RLF, CLF, + PolynomialRing, + QuadraticField) +from mjo.eja.eja_element import FiniteDimensionalEJAElement +from mjo.eja.eja_operator import FiniteDimensionalEJAOperator +from mjo.eja.eja_utils import _all2list, _mat2vec + +class FiniteDimensionalEJA(CombinatorialFreeModule): + r""" + A finite-dimensional Euclidean Jordan algebra. + + INPUT: + + - ``basis`` -- a tuple; a tuple of basis elements in "matrix + form," which must be the same form as the arguments to + ``jordan_product`` and ``inner_product``. In reality, "matrix + form" can be either vectors, matrices, or a Cartesian product + (ordered tuple) of vectors or matrices. All of these would + ideally be vector spaces in sage with no special-casing + needed; but in reality we turn vectors into column-matrices + and Cartesian products `(a,b)` into column matrices + `(a,b)^{T}` after converting `a` and `b` themselves. + + - ``jordan_product`` -- a function; afunction of two ``basis`` + elements (in matrix form) that returns their jordan product, + also in matrix form; this will be applied to ``basis`` to + compute a multiplication table for the algebra. + + - ``inner_product`` -- a function; a function of two ``basis`` + elements (in matrix form) that returns their inner + product. This will be applied to ``basis`` to compute an + inner-product table (basically a matrix) for this algebra. + + - ``field`` -- a subfield of the reals (default: ``AA``); the scalar + field for the algebra. + + - ``orthonormalize`` -- boolean (default: ``True``); whether or + not to orthonormalize the basis. Doing so is expensive and + generally rules out using the rationals as your ``field``, but + is required for spectral decompositions. + + SETUP:: -from mjo.eja.eja_operator import FiniteDimensionalEuclideanJordanAlgebraOperator + sage: from mjo.eja.eja_algebra import random_eja + + TESTS: + We should compute that an element subalgebra is associative even + if we circumvent the element method:: -class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): - @staticmethod - def __classcall_private__(cls, - field, - mult_table, - names='e', - assume_associative=False, - category=None, - rank=None, - natural_basis=None): - n = len(mult_table) - mult_table = [b.base_extend(field) for b in mult_table] - for b in mult_table: - b.set_immutable() - if not (is_Matrix(b) and b.dimensions() == (n, n)): - raise ValueError("input is not a multiplication table") - mult_table = tuple(mult_table) - - cat = FiniteDimensionalAlgebrasWithBasis(field) - cat.or_subcategory(category) - if assume_associative: - cat = cat.Associative() - - names = normalize_names(n, names) - - fda = super(FiniteDimensionalEuclideanJordanAlgebra, cls) - return fda.__classcall__(cls, - field, - mult_table, - assume_associative=assume_associative, - names=names, - category=cat, - rank=rank, - natural_basis=natural_basis) + sage: set_random_seed() + sage: J = random_eja(field=QQ,orthonormalize=False) + sage: x = J.random_element() + sage: A = x.subalgebra_generated_by(orthonormalize=False) + sage: basis = tuple(b.superalgebra_element() for b in A.basis()) + sage: J.subalgebra(basis, orthonormalize=False).is_associative() + True + """ + Element = FiniteDimensionalEJAElement def __init__(self, - field, - mult_table, - names='e', - assume_associative=False, - category=None, - rank=None, - natural_basis=None): + basis, + jordan_product, + inner_product, + field=AA, + orthonormalize=True, + associative=None, + cartesian_product=False, + check_field=True, + check_axioms=True, + prefix="b"): + + n = len(basis) + + if check_field: + if not field.is_subring(RR): + # Note: this does return true for the real algebraic + # field, the rationals, and any quadratic field where + # we've specified a real embedding. + raise ValueError("scalar field is not real") + + if check_axioms: + # Check commutativity of the Jordan and inner-products. + # This has to be done before we build the multiplication + # and inner-product tables/matrices, because we take + # advantage of symmetry in the process. + if not all( jordan_product(bi,bj) == jordan_product(bj,bi) + for bi in basis + for bj in basis ): + raise ValueError("Jordan product is not commutative") + + if not all( inner_product(bi,bj) == inner_product(bj,bi) + for bi in basis + for bj in basis ): + raise ValueError("inner-product is not commutative") + + + category = MagmaticAlgebras(field).FiniteDimensional() + category = category.WithBasis().Unital().Commutative() + + if n <= 1: + # All zero- and one-dimensional algebras are just the real + # numbers with (some positive multiples of) the usual + # multiplication as its Jordan and inner-product. + associative = True + if associative is None: + # We should figure it out. As with check_axioms, we have to do + # this without the help of the _jordan_product_is_associative() + # method because we need to know the category before we + # initialize the algebra. + associative = all( jordan_product(jordan_product(bi,bj),bk) + == + jordan_product(bi,jordan_product(bj,bk)) + for bi in basis + for bj in basis + for bk in basis) + + if associative: + # Element subalgebras can take advantage of this. + category = category.Associative() + if cartesian_product: + # Use join() here because otherwise we only get the + # "Cartesian product of..." and not the things themselves. + category = category.join([category, + category.CartesianProducts()]) + + # Call the superclass constructor so that we can use its from_vector() + # method to build our multiplication table. + CombinatorialFreeModule.__init__(self, + field, + range(n), + prefix=prefix, + category=category, + bracket=False) + + # Now comes all of the hard work. We'll be constructing an + # ambient vector space V that our (vectorized) basis lives in, + # as well as a subspace W of V spanned by those (vectorized) + # basis elements. The W-coordinates are the coefficients that + # we see in things like x = 1*b1 + 2*b2. + vector_basis = basis + + degree = 0 + if n > 0: + degree = len(_all2list(basis[0])) + + # Build an ambient space that fits our matrix basis when + # written out as "long vectors." + V = VectorSpace(field, degree) + + # The matrix that will hole the orthonormal -> unorthonormal + # coordinate transformation. + self._deortho_matrix = None + + if orthonormalize: + # Save a copy of the un-orthonormalized basis for later. + # Convert it to ambient V (vector) coordinates while we're + # at it, because we'd have to do it later anyway. + deortho_vector_basis = tuple( V(_all2list(b)) for b in basis ) + + from mjo.eja.eja_utils import gram_schmidt + basis = tuple(gram_schmidt(basis, inner_product)) + + # Save the (possibly orthonormalized) matrix basis for + # later... + self._matrix_basis = basis + + # Now create the vector space for the algebra, which will have + # its own set of non-ambient coordinates (in terms of the + # supplied basis). + vector_basis = tuple( V(_all2list(b)) for b in basis ) + W = V.span_of_basis( vector_basis, check=check_axioms) + + if orthonormalize: + # Now "W" is the vector space of our algebra coordinates. The + # variables "X1", "X2",... refer to the entries of vectors in + # W. Thus to convert back and forth between the orthonormal + # coordinates and the given ones, we need to stick the original + # basis in W. + U = V.span_of_basis( deortho_vector_basis, check=check_axioms) + self._deortho_matrix = matrix( U.coordinate_vector(q) + for q in vector_basis ) + + + # Now we actually compute the multiplication and inner-product + # tables/matrices using the possibly-orthonormalized basis. + self._inner_product_matrix = matrix.identity(field, n) + self._multiplication_table = [ [0 for j in range(i+1)] + for i in range(n) ] + + # Note: the Jordan and inner-products are defined in terms + # of the ambient basis. It's important that their arguments + # are in ambient coordinates as well. + for i in range(n): + for j in range(i+1): + # ortho basis w.r.t. ambient coords + q_i = basis[i] + q_j = basis[j] + + # The jordan product returns a matrixy answer, so we + # have to convert it to the algebra coordinates. + elt = jordan_product(q_i, q_j) + elt = W.coordinate_vector(V(_all2list(elt))) + self._multiplication_table[i][j] = self.from_vector(elt) + + if not orthonormalize: + # If we're orthonormalizing the basis with respect + # to an inner-product, then the inner-product + # matrix with respect to the resulting basis is + # just going to be the identity. + ip = inner_product(q_i, q_j) + self._inner_product_matrix[i,j] = ip + self._inner_product_matrix[j,i] = ip + + self._inner_product_matrix._cache = {'hermitian': True} + self._inner_product_matrix.set_immutable() + + if check_axioms: + if not self._is_jordanian(): + raise ValueError("Jordan identity does not hold") + if not self._inner_product_is_associative(): + raise ValueError("inner product is not associative") + + + def _coerce_map_from_base_ring(self): + """ + Disable the map from the base ring into the algebra. + + Performing a nonsense conversion like this automatically + is counterpedagogical. The fallback is to try the usual + element constructor, which should also fail. + + SETUP:: + + sage: from mjo.eja.eja_algebra import random_eja + + TESTS:: + + sage: set_random_seed() + sage: J = random_eja() + sage: J(1) + Traceback (most recent call last): + ... + ValueError: not an element of this algebra + """ + return None + + + def product_on_basis(self, i, j): + r""" + Returns the Jordan product of the `i` and `j`th basis elements. + + This completely defines the Jordan product on the algebra, and + is used direclty by our superclass machinery to implement + :meth:`product`. + SETUP:: sage: from mjo.eja.eja_algebra import random_eja + TESTS:: + + sage: set_random_seed() + sage: J = random_eja() + sage: n = J.dimension() + sage: bi = J.zero() + sage: bj = J.zero() + sage: bi_bj = J.zero()*J.zero() + sage: if n > 0: + ....: i = ZZ.random_element(n) + ....: j = ZZ.random_element(n) + ....: bi = J.monomial(i) + ....: bj = J.monomial(j) + ....: bi_bj = J.product_on_basis(i,j) + sage: bi*bj == bi_bj + True + + """ + # We only stored the lower-triangular portion of the + # multiplication table. + if j <= i: + return self._multiplication_table[i][j] + else: + return self._multiplication_table[j][i] + + def inner_product(self, x, y): + """ + The inner product associated with this Euclidean Jordan algebra. + + Defaults to the trace inner product, but can be overridden by + subclasses if they are sure that the necessary properties are + satisfied. + + SETUP:: + + sage: from mjo.eja.eja_algebra import (random_eja, + ....: HadamardEJA, + ....: BilinearFormEJA) + EXAMPLES: - By definition, Jordan multiplication commutes:: + Our inner product is "associative," which means the following for + a symmetric bilinear form:: sage: set_random_seed() sage: J = random_eja() + sage: x,y,z = J.random_elements(3) + sage: (x*y).inner_product(z) == y.inner_product(x*z) + True + + TESTS: + + Ensure that this is the usual inner product for the algebras + over `R^n`:: + + sage: set_random_seed() + sage: J = HadamardEJA.random_instance() + sage: x,y = J.random_elements(2) + sage: actual = x.inner_product(y) + sage: expected = x.to_vector().inner_product(y.to_vector()) + sage: actual == expected + True + + Ensure that this is one-half of the trace inner-product in a + BilinearFormEJA that isn't just the reals (when ``n`` isn't + one). This is in Faraut and Koranyi, and also my "On the + symmetry..." paper:: + + sage: set_random_seed() + sage: J = BilinearFormEJA.random_instance() + sage: n = J.dimension() sage: x = J.random_element() sage: y = J.random_element() - sage: x*y == y*x + sage: (n == 1) or (x.inner_product(y) == (x*y).trace()/2) True """ - self._rank = rank - self._natural_basis = natural_basis - self._multiplication_table = mult_table - fda = super(FiniteDimensionalEuclideanJordanAlgebra, self) - fda.__init__(field, - mult_table, - names=names, - category=category) + B = self._inner_product_matrix + return (B*x.to_vector()).inner_product(y.to_vector()) - def _repr_(self): - """ - Return a string representation of ``self``. + def is_associative(self): + r""" + Return whether or not this algebra's Jordan product is associative. SETUP:: - sage: from mjo.eja.eja_algebra import JordanSpinEJA - - TESTS: + sage: from mjo.eja.eja_algebra import ComplexHermitianEJA - Ensure that it says what we think it says:: + EXAMPLES:: - sage: JordanSpinEJA(2, field=QQ) - Euclidean Jordan algebra of degree 2 over Rational Field - sage: JordanSpinEJA(3, field=RDF) - Euclidean Jordan algebra of degree 3 over Real Double Field + sage: J = ComplexHermitianEJA(3, field=QQ, orthonormalize=False) + sage: J.is_associative() + False + sage: x = sum(J.gens()) + sage: A = x.subalgebra_generated_by(orthonormalize=False) + sage: A.is_associative() + True """ - fmt = "Euclidean Jordan algebra of degree {} over {}" - return fmt.format(self.degree(), self.base_ring()) + return "Associative" in self.category().axioms() + def _is_commutative(self): + r""" + Whether or not this algebra's multiplication table is commutative. - def _a_regular_element(self): + This method should of course always return ``True``, unless + this algebra was constructed with ``check_axioms=False`` and + passed an invalid multiplication table. + """ + return all( x*y == y*x for x in self.gens() for y in self.gens() ) + + def _is_jordanian(self): + r""" + Whether or not this algebra's multiplication table respects the + Jordan identity `(x^{2})(xy) = x(x^{2}y)`. + + We only check one arrangement of `x` and `y`, so for a + ``True`` result to be truly true, you should also check + :meth:`_is_commutative`. This method should of course always + return ``True``, unless this algebra was constructed with + ``check_axioms=False`` and passed an invalid multiplication table. """ - Guess a regular element. Needed to compute the basis for our - characteristic polynomial coefficients. + return all( (self.monomial(i)**2)*(self.monomial(i)*self.monomial(j)) + == + (self.monomial(i))*((self.monomial(i)**2)*self.monomial(j)) + for i in range(self.dimension()) + for j in range(self.dimension()) ) + + def _jordan_product_is_associative(self): + r""" + Return whether or not this algebra's Jordan product is + associative; that is, whether or not `x*(y*z) = (x*y)*z` + for all `x,y,x`. + + This method should agree with :meth:`is_associative` unless + you lied about the value of the ``associative`` parameter + when you constructed the algebra. SETUP:: - sage: from mjo.eja.eja_algebra import random_eja + sage: from mjo.eja.eja_algebra import (random_eja, + ....: RealSymmetricEJA, + ....: ComplexHermitianEJA, + ....: QuaternionHermitianEJA) + + EXAMPLES:: + + sage: J = RealSymmetricEJA(4, orthonormalize=False) + sage: J._jordan_product_is_associative() + False + sage: x = sum(J.gens()) + sage: A = x.subalgebra_generated_by() + sage: A._jordan_product_is_associative() + True + + :: + + sage: J = ComplexHermitianEJA(2,field=QQ,orthonormalize=False) + sage: J._jordan_product_is_associative() + False + sage: x = sum(J.gens()) + sage: A = x.subalgebra_generated_by(orthonormalize=False) + sage: A._jordan_product_is_associative() + True + + :: + + sage: J = QuaternionHermitianEJA(2) + sage: J._jordan_product_is_associative() + False + sage: x = sum(J.gens()) + sage: A = x.subalgebra_generated_by() + sage: A._jordan_product_is_associative() + True TESTS: - Ensure that this hacky method succeeds for every algebra that we - know how to construct:: + The values we've presupplied to the constructors agree with + the computation:: sage: set_random_seed() sage: J = random_eja() - sage: J._a_regular_element().is_regular() + sage: J.is_associative() == J._jordan_product_is_associative() True """ - gs = self.gens() - z = self.sum( (i+1)*gs[i] for i in range(len(gs)) ) - if not z.is_regular(): - raise ValueError("don't know a regular element") - return z + R = self.base_ring() + + # Used to check whether or not something is zero. + epsilon = R.zero() + if not R.is_exact(): + # I don't know of any examples that make this magnitude + # necessary because I don't know how to make an + # associative algebra when the element subalgebra + # construction is unreliable (as it is over RDF; we can't + # find the degree of an element because we can't compute + # the rank of a matrix). But even multiplication of floats + # is non-associative, so *some* epsilon is needed... let's + # just take the one from _inner_product_is_associative? + epsilon = 1e-15 + + for i in range(self.dimension()): + for j in range(self.dimension()): + for k in range(self.dimension()): + x = self.monomial(i) + y = self.monomial(j) + z = self.monomial(k) + diff = (x*y)*z - x*(y*z) + + if diff.norm() > epsilon: + return False + + return True + + def _inner_product_is_associative(self): + r""" + Return whether or not this algebra's inner product `B` is + associative; that is, whether or not `B(xy,z) = B(x,yz)`. + + This method should of course always return ``True``, unless + this algebra was constructed with ``check_axioms=False`` and + passed an invalid Jordan or inner-product. + """ + R = self.base_ring() + # Used to check whether or not something is zero. + epsilon = R.zero() + if not R.is_exact(): + # This choice is sufficient to allow the construction of + # QuaternionHermitianEJA(2, field=RDF) with check_axioms=True. + epsilon = 1e-15 - @cached_method - def _charpoly_basis_space(self): - """ - Return the vector space spanned by the basis used in our - characteristic polynomial coefficients. This is used not only to - compute those coefficients, but also any time we need to - evaluate the coefficients (like when we compute the trace or - determinant). + for i in range(self.dimension()): + for j in range(self.dimension()): + for k in range(self.dimension()): + x = self.monomial(i) + y = self.monomial(j) + z = self.monomial(k) + diff = (x*y).inner_product(z) - x.inner_product(y*z) + + if diff.abs() > epsilon: + return False + + return True + + def _element_constructor_(self, elt): """ - z = self._a_regular_element() - V = self.vector_space() - V1 = V.span_of_basis( (z**k).vector() for k in range(self.rank()) ) - b = (V1.basis() + V1.complement().basis()) - return V.span_of_basis(b) + Construct an element of this algebra from its vector or matrix + representation. + This gets called only after the parent element _call_ method + fails to find a coercion for the argument. - @cached_method - def _charpoly_coeff(self, i): - """ - Return the coefficient polynomial "a_{i}" of this algebra's - general characteristic polynomial. - - Having this be a separate cached method lets us compute and - store the trace/determinant (a_{r-1} and a_{0} respectively) - separate from the entire characteristic polynomial. - """ - (A_of_x, x, xr, detA) = self._charpoly_matrix_system() - R = A_of_x.base_ring() - if i >= self.rank(): - # Guaranteed by theory - return R.zero() - - # Danger: the in-place modification is done for performance - # reasons (reconstructing a matrix with huge polynomial - # entries is slow), but I don't know how cached_method works, - # so it's highly possible that we're modifying some global - # list variable by reference, here. In other words, you - # probably shouldn't call this method twice on the same - # algebra, at the same time, in two threads - Ai_orig = A_of_x.column(i) - A_of_x.set_column(i,xr) - numerator = A_of_x.det() - A_of_x.set_column(i,Ai_orig) - - # We're relying on the theory here to ensure that each a_i is - # indeed back in R, and the added negative signs are to make - # the whole charpoly expression sum to zero. - return R(-numerator/detA) + SETUP:: + sage: from mjo.eja.eja_algebra import (random_eja, + ....: JordanSpinEJA, + ....: HadamardEJA, + ....: RealSymmetricEJA) - @cached_method - def _charpoly_matrix_system(self): - """ - Compute the matrix whose entries A_ij are polynomials in - X1,...,XN, the vector ``x`` of variables X1,...,XN, the vector - corresponding to `x^r` and the determinent of the matrix A = - [A_ij]. In other words, all of the fixed (cachable) data needed - to compute the coefficients of the characteristic polynomial. - """ - r = self.rank() - n = self.dimension() + EXAMPLES: + + The identity in `S^n` is converted to the identity in the EJA:: + + sage: J = RealSymmetricEJA(3) + sage: I = matrix.identity(QQ,3) + sage: J(I) == J.one() + True + + This skew-symmetric matrix can't be represented in the EJA:: + + sage: J = RealSymmetricEJA(3) + sage: A = matrix(QQ,3, lambda i,j: i-j) + sage: J(A) + Traceback (most recent call last): + ... + ValueError: not an element of this algebra + + Tuples work as well, provided that the matrix basis for the + algebra consists of them:: + + sage: J1 = HadamardEJA(3) + sage: J2 = RealSymmetricEJA(2) + sage: J = cartesian_product([J1,J2]) + sage: J( (J1.matrix_basis()[1], J2.matrix_basis()[2]) ) + b1 + b5 - # Construct a new algebra over a multivariate polynomial ring... - names = ['X' + str(i) for i in range(1,n+1)] - R = PolynomialRing(self.base_ring(), names) - J = FiniteDimensionalEuclideanJordanAlgebra(R, - self._multiplication_table, - rank=r) + TESTS: - idmat = matrix.identity(J.base_ring(), n) + Ensure that we can convert any element back and forth + faithfully between its matrix and algebra representations:: - W = self._charpoly_basis_space() - W = W.change_ring(R.fraction_field()) + sage: set_random_seed() + sage: J = random_eja() + sage: x = J.random_element() + sage: J(x.to_matrix()) == x + True - # Starting with the standard coordinates x = (X1,X2,...,Xn) - # and then converting the entries to W-coordinates allows us - # to pass in the standard coordinates to the charpoly and get - # back the right answer. Specifically, with x = (X1,X2,...,Xn), - # we have + We cannot coerce elements between algebras just because their + matrix representations are compatible:: + + sage: J1 = HadamardEJA(3) + sage: J2 = JordanSpinEJA(3) + sage: J2(J1.one()) + Traceback (most recent call last): + ... + ValueError: not an element of this algebra + sage: J1(J2.zero()) + Traceback (most recent call last): + ... + ValueError: not an element of this algebra + """ + msg = "not an element of this algebra" + if elt in self.base_ring(): + # Ensure that no base ring -> algebra coercion is performed + # by this method. There's some stupidity in sage that would + # otherwise propagate to this method; for example, sage thinks + # that the integer 3 belongs to the space of 2-by-2 matrices. + raise ValueError(msg) + + try: + # Try to convert a vector into a column-matrix... + elt = elt.column() + except (AttributeError, TypeError): + # and ignore failure, because we weren't really expecting + # a vector as an argument anyway. + pass + + if elt not in self.matrix_space(): + raise ValueError(msg) + + # Thanks for nothing! Matrix spaces aren't vector spaces in + # Sage, so we have to figure out its matrix-basis coordinates + # ourselves. We use the basis space's ring instead of the + # element's ring because the basis space might be an algebraic + # closure whereas the base ring of the 3-by-3 identity matrix + # could be QQ instead of QQbar. # - # W.coordinates(x^2) eval'd at (standard z-coords) - # = - # W-coords of (z^2) - # = - # W-coords of (standard coords of x^2 eval'd at std-coords of z) + # And, we also have to handle Cartesian product bases (when + # the matrix basis consists of tuples) here. The "good news" + # is that we're already converting everything to long vectors, + # and that strategy works for tuples as well. # - # We want the middle equivalent thing in our matrix, but use - # the first equivalent thing instead so that we can pass in - # standard coordinates. - x = J(W(R.gens())) - l1 = [matrix.column(W.coordinates((x**k).vector())) for k in range(r)] - l2 = [idmat.column(k-1).column() for k in range(r+1, n+1)] - A_of_x = matrix.block(R, 1, n, (l1 + l2)) - xr = W.coordinates((x**r).vector()) - return (A_of_x, x, xr, A_of_x.det()) + # We pass check=False because the matrix basis is "guaranteed" + # to be linearly independent... right? Ha ha. + elt = _all2list(elt) + V = VectorSpace(self.base_ring(), len(elt)) + W = V.span_of_basis( (V(_all2list(s)) for s in self.matrix_basis()), + check=False) + + try: + coords = W.coordinate_vector(V(elt)) + except ArithmeticError: # vector is not in free module + raise ValueError(msg) + + return self.from_vector(coords) + + def _repr_(self): + """ + Return a string representation of ``self``. + + SETUP:: + + sage: from mjo.eja.eja_algebra import JordanSpinEJA + + TESTS: + + Ensure that it says what we think it says:: + + sage: JordanSpinEJA(2, field=AA) + Euclidean Jordan algebra of dimension 2 over Algebraic Real Field + sage: JordanSpinEJA(3, field=RDF) + Euclidean Jordan algebra of dimension 3 over Real Double Field + + """ + fmt = "Euclidean Jordan algebra of dimension {} over {}" + return fmt.format(self.dimension(), self.base_ring()) @cached_method - def characteristic_polynomial(self): + def characteristic_polynomial_of(self): """ - Return a characteristic polynomial that works for all elements - of this algebra. + Return the algebra's "characteristic polynomial of" function, + which is itself a multivariate polynomial that, when evaluated + at the coordinates of some algebra element, returns that + element's characteristic polynomial. The resulting polynomial has `n+1` variables, where `n` is the dimension of this algebra. The first `n` variables correspond to @@ -260,7 +731,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): SETUP:: - sage: from mjo.eja.eja_algebra import JordanSpinEJA + sage: from mjo.eja.eja_algebra import JordanSpinEJA, TrivialEJA EXAMPLES: @@ -268,42 +739,64 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): Alizadeh, Example 11.11:: sage: J = JordanSpinEJA(3) - sage: p = J.characteristic_polynomial(); p + sage: p = J.characteristic_polynomial_of(); p X1^2 - X2^2 - X3^2 + (-2*t)*X1 + t^2 - sage: xvec = J.one().vector() + sage: xvec = J.one().to_vector() sage: p(*xvec) t^2 - 2*t + 1 + By definition, the characteristic polynomial is a monic + degree-zero polynomial in a rank-zero algebra. Note that + Cayley-Hamilton is indeed satisfied since the polynomial + ``1`` evaluates to the identity element of the algebra on + any argument:: + + sage: J = TrivialEJA() + sage: J.characteristic_polynomial_of() + 1 + """ r = self.rank() n = self.dimension() - # The list of coefficient polynomials a_1, a_2, ..., a_n. - a = [ self._charpoly_coeff(i) for i in range(n) ] + # The list of coefficient polynomials a_0, a_1, a_2, ..., a_(r-1). + a = self._charpoly_coefficients() # We go to a bit of trouble here to reorder the # indeterminates, so that it's easier to evaluate the # characteristic polynomial at x's coordinates and get back # something in terms of t, which is what we want. - R = a[0].parent() S = PolynomialRing(self.base_ring(),'t') t = S.gen(0) - S = PolynomialRing(S, R.variable_names()) - t = S(t) - - # Note: all entries past the rth should be zero. The - # coefficient of the highest power (x^r) is 1, but it doesn't - # appear in the solution vector which contains coefficients - # for the other powers (to make them sum to x^r). - if (r < n): - a[r] = 1 # corresponds to x^r - else: - # When the rank is equal to the dimension, trying to - # assign a[r] goes out-of-bounds. - a.append(1) # corresponds to x^r + if r > 0: + R = a[0].parent() + S = PolynomialRing(S, R.variable_names()) + t = S(t) + + return (t**r + sum( a[k]*(t**k) for k in range(r) )) + + def coordinate_polynomial_ring(self): + r""" + The multivariate polynomial ring in which this algebra's + :meth:`characteristic_polynomial_of` lives. + + SETUP:: - return sum( a[k]*(t**k) for k in range(len(a)) ) + sage: from mjo.eja.eja_algebra import (HadamardEJA, + ....: RealSymmetricEJA) + + EXAMPLES:: + + sage: J = HadamardEJA(2) + sage: J.coordinate_polynomial_ring() + Multivariate Polynomial Ring in X1, X2... + sage: J = RealSymmetricEJA(3,field=QQ,orthonormalize=False) + sage: J.coordinate_polynomial_ring() + Multivariate Polynomial Ring in X1, X2, X3, X4, X5, X6... + """ + var_names = tuple( "X%d" % z for z in range(1, self.dimension()+1) ) + return PolynomialRing(self.base_ring(), var_names) def inner_product(self, x, y): """ @@ -315,39 +808,144 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): SETUP:: - sage: from mjo.eja.eja_algebra import random_eja + sage: from mjo.eja.eja_algebra import (random_eja, + ....: HadamardEJA, + ....: BilinearFormEJA) EXAMPLES: - The inner product must satisfy its axiom for this algebra to truly - be a Euclidean Jordan Algebra:: + Our inner product is "associative," which means the following for + a symmetric bilinear form:: sage: set_random_seed() sage: J = random_eja() + sage: x,y,z = J.random_elements(3) + sage: (x*y).inner_product(z) == y.inner_product(x*z) + True + + TESTS: + + Ensure that this is the usual inner product for the algebras + over `R^n`:: + + sage: set_random_seed() + sage: J = HadamardEJA.random_instance() + sage: x,y = J.random_elements(2) + sage: actual = x.inner_product(y) + sage: expected = x.to_vector().inner_product(y.to_vector()) + sage: actual == expected + True + + Ensure that this is one-half of the trace inner-product in a + BilinearFormEJA that isn't just the reals (when ``n`` isn't + one). This is in Faraut and Koranyi, and also my "On the + symmetry..." paper:: + + sage: set_random_seed() + sage: J = BilinearFormEJA.random_instance() + sage: n = J.dimension() sage: x = J.random_element() sage: y = J.random_element() - sage: z = J.random_element() - sage: (x*y).inner_product(z) == y.inner_product(x*z) + sage: (n == 1) or (x.inner_product(y) == (x*y).trace()/2) + True + """ + B = self._inner_product_matrix + return (B*x.to_vector()).inner_product(y.to_vector()) + + + def is_trivial(self): + """ + Return whether or not this algebra is trivial. + + A trivial algebra contains only the zero element. + + SETUP:: + + sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA, + ....: TrivialEJA) + + EXAMPLES:: + + sage: J = ComplexHermitianEJA(3) + sage: J.is_trivial() + False + + :: + + sage: J = TrivialEJA() + sage: J.is_trivial() True """ - if (not x in self) or (not y in self): - raise TypeError("arguments must live in this algebra") - return x.trace_inner_product(y) + return self.dimension() == 0 + + + def multiplication_table(self): + """ + Return a visual representation of this algebra's multiplication + table (on basis elements). + + SETUP:: + + sage: from mjo.eja.eja_algebra import JordanSpinEJA + EXAMPLES:: + + sage: J = JordanSpinEJA(4) + sage: J.multiplication_table() + +----++----+----+----+----+ + | * || b0 | b1 | b2 | b3 | + +====++====+====+====+====+ + | b0 || b0 | b1 | b2 | b3 | + +----++----+----+----+----+ + | b1 || b1 | b0 | 0 | 0 | + +----++----+----+----+----+ + | b2 || b2 | 0 | b0 | 0 | + +----++----+----+----+----+ + | b3 || b3 | 0 | 0 | b0 | + +----++----+----+----+----+ - def natural_basis(self): """ - Return a more-natural representation of this algebra's basis. + n = self.dimension() + # Prepend the header row. + M = [["*"] + list(self.gens())] + + # And to each subsequent row, prepend an entry that belongs to + # the left-side "header column." + M += [ [self.monomial(i)] + [ self.monomial(i)*self.monomial(j) + for j in range(n) ] + for i in range(n) ] - Every finite-dimensional Euclidean Jordan Algebra is a direct - sum of five simple algebras, four of which comprise Hermitian - matrices. This method returns the original "natural" basis - for our underlying vector space. (Typically, the natural basis - is used to construct the multiplication table in the first place.) + return table(M, header_row=True, header_column=True, frame=True) - Note that this will always return a matrix. The standard basis - in `R^n` will be returned as `n`-by-`1` column matrices. + + def matrix_basis(self): + """ + Return an (often more natural) representation of this algebras + basis as an ordered tuple of matrices. + + Every finite-dimensional Euclidean Jordan Algebra is a, up to + Jordan isomorphism, a direct sum of five simple + algebras---four of which comprise Hermitian matrices. And the + last type of algebra can of course be thought of as `n`-by-`1` + column matrices (ambiguusly called column vectors) to avoid + special cases. As a result, matrices (and column vectors) are + a natural representation format for Euclidean Jordan algebra + elements. + + But, when we construct an algebra from a basis of matrices, + those matrix representations are lost in favor of coordinate + vectors *with respect to* that basis. We could eventually + convert back if we tried hard enough, but having the original + representations handy is valuable enough that we simply store + them and return them from this method. + + Why implement this for non-matrix algebras? Avoiding special + cases for the :class:`BilinearFormEJA` pays with simplicity in + its own right. But mainly, we would like to be able to assume + that elements of a :class:`CartesianProductEJA` can be displayed + nicely, without having to have special classes for direct sums + one of whose components was a matrix algebra. SETUP:: @@ -358,1968 +956,2296 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): sage: J = RealSymmetricEJA(2) sage: J.basis() - Family (e0, e1, e2) - sage: J.natural_basis() + Finite family {0: b0, 1: b1, 2: b2} + sage: J.matrix_basis() ( - [1 0] [0 1] [0 0] - [0 0], [1 0], [0 1] + [1 0] [ 0 0.7071067811865475?] [0 0] + [0 0], [0.7071067811865475? 0], [0 1] ) :: sage: J = JordanSpinEJA(2) sage: J.basis() - Family (e0, e1) - sage: J.natural_basis() + Finite family {0: b0, 1: b1} + sage: J.matrix_basis() ( [1] [0] [0], [1] ) - """ - if self._natural_basis is None: - return tuple( b.vector().column() for b in self.basis() ) - else: - return self._natural_basis + return self._matrix_basis - def rank(self): - """ - Return the rank of this EJA. + def matrix_space(self): """ - if self._rank is None: - raise ValueError("no rank specified at genesis") - else: - return self._rank + Return the matrix space in which this algebra's elements live, if + we think of them as matrices (including column vectors of the + appropriate size). - - def vector_space(self): - """ - Return the vector space that underlies this algebra. + "By default" this will be an `n`-by-`1` column-matrix space, + except when the algebra is trivial. There it's `n`-by-`n` + (where `n` is zero), to ensure that two elements of the matrix + space (empty matrices) can be multiplied. For algebras of + matrices, this returns the space in which their + real embeddings live. SETUP:: - sage: from mjo.eja.eja_algebra import RealSymmetricEJA + sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA, + ....: JordanSpinEJA, + ....: QuaternionHermitianEJA, + ....: TrivialEJA) - EXAMPLES:: + EXAMPLES: - sage: J = RealSymmetricEJA(2) - sage: J.vector_space() - Vector space of dimension 3 over Rational Field + By default, the matrix representation is just a column-matrix + equivalent to the vector representation:: + + sage: J = JordanSpinEJA(3) + sage: J.matrix_space() + Full MatrixSpace of 3 by 1 dense matrices over Algebraic + Real Field + + The matrix representation in the trivial algebra is + zero-by-zero instead of the usual `n`-by-one:: + + sage: J = TrivialEJA() + sage: J.matrix_space() + Full MatrixSpace of 0 by 0 dense matrices over Algebraic + Real Field + + The matrix space for complex/quaternion Hermitian matrix EJA + is the space in which their real-embeddings live, not the + original complex/quaternion matrix space:: + + sage: J = ComplexHermitianEJA(2,field=QQ,orthonormalize=False) + sage: J.matrix_space() + Full MatrixSpace of 4 by 4 dense matrices over Rational Field + sage: J = QuaternionHermitianEJA(1,field=QQ,orthonormalize=False) + sage: J.matrix_space() + Module of 1 by 1 matrices with entries in Quaternion + Algebra (-1, -1) with base ring Rational Field over + the scalar ring Rational Field """ - return self.zero().vector().parent().ambient_vector_space() + if self.is_trivial(): + return MatrixSpace(self.base_ring(), 0) + else: + return self.matrix_basis()[0].parent() - class Element(FiniteDimensionalAlgebraElement): - """ - An element of a Euclidean Jordan algebra. + @cached_method + def one(self): """ + Return the unit element of this algebra. - def __dir__(self): - """ - Oh man, I should not be doing this. This hides the "disabled" - methods ``left_matrix`` and ``matrix`` from introspection; - in particular it removes them from tab-completion. - """ - return filter(lambda s: s not in ['left_matrix', 'matrix'], - dir(self.__class__) ) + SETUP:: + sage: from mjo.eja.eja_algebra import (HadamardEJA, + ....: random_eja) - def __init__(self, A, elt=None): - """ + EXAMPLES: - SETUP:: + We can compute unit element in the Hadamard EJA:: - sage: from mjo.eja.eja_algebra import RealSymmetricEJA + sage: J = HadamardEJA(5) + sage: J.one() + b0 + b1 + b2 + b3 + b4 - EXAMPLES: + The unit element in the Hadamard EJA is inherited in the + subalgebras generated by its elements:: - The identity in `S^n` is converted to the identity in the EJA:: + sage: J = HadamardEJA(5) + sage: J.one() + b0 + b1 + b2 + b3 + b4 + sage: x = sum(J.gens()) + sage: A = x.subalgebra_generated_by(orthonormalize=False) + sage: A.one() + c0 + sage: A.one().superalgebra_element() + b0 + b1 + b2 + b3 + b4 - sage: J = RealSymmetricEJA(3) - sage: I = matrix.identity(QQ,3) - sage: J(I) == J.one() - True + TESTS: - This skew-symmetric matrix can't be represented in the EJA:: + The identity element acts like the identity, regardless of + whether or not we orthonormalize:: - sage: J = RealSymmetricEJA(3) - sage: A = matrix(QQ,3, lambda i,j: i-j) - sage: J(A) - Traceback (most recent call last): - ... - ArithmeticError: vector is not in free module + sage: set_random_seed() + sage: J = random_eja() + sage: x = J.random_element() + sage: J.one()*x == x and x*J.one() == x + True + sage: A = x.subalgebra_generated_by() + sage: y = A.random_element() + sage: A.one()*y == y and y*A.one() == y + True - """ - # Goal: if we're given a matrix, and if it lives in our - # parent algebra's "natural ambient space," convert it - # into an algebra element. - # - # The catch is, we make a recursive call after converting - # the given matrix into a vector that lives in the algebra. - # This we need to try the parent class initializer first, - # to avoid recursing forever if we're given something that - # already fits into the algebra, but also happens to live - # in the parent's "natural ambient space" (this happens with - # vectors in R^n). - try: - FiniteDimensionalAlgebraElement.__init__(self, A, elt) - except ValueError: - natural_basis = A.natural_basis() - if elt in natural_basis[0].matrix_space(): - # Thanks for nothing! Matrix spaces aren't vector - # spaces in Sage, so we have to figure out its - # natural-basis coordinates ourselves. - V = VectorSpace(elt.base_ring(), elt.nrows()**2) - W = V.span( _mat2vec(s) for s in natural_basis ) - coords = W.coordinates(_mat2vec(elt)) - FiniteDimensionalAlgebraElement.__init__(self, A, coords) - - def __pow__(self, n): - """ - Return ``self`` raised to the power ``n``. - - Jordan algebras are always power-associative; see for - example Faraut and Koranyi, Proposition II.1.2 (ii). - - .. WARNING: - - We have to override this because our superclass uses row vectors - instead of column vectors! We, on the other hand, assume column - vectors everywhere. - - SETUP:: - - sage: from mjo.eja.eja_algebra import random_eja - - EXAMPLES:: - - sage: set_random_seed() - sage: x = random_eja().random_element() - sage: x.operator()(x) == (x^2) - True - - A few examples of power-associativity:: - - sage: set_random_seed() - sage: x = random_eja().random_element() - sage: x*(x*x)*(x*x) == x^5 - True - sage: (x*x)*(x*x*x) == x^5 - True - - We also know that powers operator-commute (Koecher, Chapter - III, Corollary 1):: - - sage: set_random_seed() - sage: x = random_eja().random_element() - sage: m = ZZ.random_element(0,10) - sage: n = ZZ.random_element(0,10) - sage: Lxm = (x^m).operator() - sage: Lxn = (x^n).operator() - sage: Lxm*Lxn == Lxn*Lxm - True - - """ - if n == 0: - return self.parent().one() - elif n == 1: - return self - else: - return (self.operator()**(n-1))(self) + :: + sage: set_random_seed() + sage: J = random_eja(field=QQ, orthonormalize=False) + sage: x = J.random_element() + sage: J.one()*x == x and x*J.one() == x + True + sage: A = x.subalgebra_generated_by(orthonormalize=False) + sage: y = A.random_element() + sage: A.one()*y == y and y*A.one() == y + True - def apply_univariate_polynomial(self, p): - """ - Apply the univariate polynomial ``p`` to this element. + The matrix of the unit element's operator is the identity, + regardless of the base field and whether or not we + orthonormalize:: - A priori, SageMath won't allow us to apply a univariate - polynomial to an element of an EJA, because we don't know - that EJAs are rings (they are usually not associative). Of - course, we know that EJAs are power-associative, so the - operation is ultimately kosher. This function sidesteps - the CAS to get the answer we want and expect. + sage: set_random_seed() + sage: J = random_eja() + sage: actual = J.one().operator().matrix() + sage: expected = matrix.identity(J.base_ring(), J.dimension()) + sage: actual == expected + True + sage: x = J.random_element() + sage: A = x.subalgebra_generated_by() + sage: actual = A.one().operator().matrix() + sage: expected = matrix.identity(A.base_ring(), A.dimension()) + sage: actual == expected + True - SETUP:: + :: - sage: from mjo.eja.eja_algebra import (RealCartesianProductEJA, - ....: random_eja) + sage: set_random_seed() + sage: J = random_eja(field=QQ, orthonormalize=False) + sage: actual = J.one().operator().matrix() + sage: expected = matrix.identity(J.base_ring(), J.dimension()) + sage: actual == expected + True + sage: x = J.random_element() + sage: A = x.subalgebra_generated_by(orthonormalize=False) + sage: actual = A.one().operator().matrix() + sage: expected = matrix.identity(A.base_ring(), A.dimension()) + sage: actual == expected + True - EXAMPLES:: + Ensure that the cached unit element (often precomputed by + hand) agrees with the computed one:: - sage: R = PolynomialRing(QQ, 't') - sage: t = R.gen(0) - sage: p = t^4 - t^3 + 5*t - 2 - sage: J = RealCartesianProductEJA(5) - sage: J.one().apply_univariate_polynomial(p) == 3*J.one() - True + sage: set_random_seed() + sage: J = random_eja() + sage: cached = J.one() + sage: J.one.clear_cache() + sage: J.one() == cached + True - TESTS: + :: - We should always get back an element of the algebra:: + sage: set_random_seed() + sage: J = random_eja(field=QQ, orthonormalize=False) + sage: cached = J.one() + sage: J.one.clear_cache() + sage: J.one() == cached + True - sage: set_random_seed() - sage: p = PolynomialRing(QQ, 't').random_element() - sage: J = random_eja() - sage: x = J.random_element() - sage: x.apply_univariate_polynomial(p) in J - True + """ + # We can brute-force compute the matrices of the operators + # that correspond to the basis elements of this algebra. + # If some linear combination of those basis elements is the + # algebra identity, then the same linear combination of + # their matrices has to be the identity matrix. + # + # Of course, matrices aren't vectors in sage, so we have to + # appeal to the "long vectors" isometry. + oper_vecs = [ _mat2vec(g.operator().matrix()) for g in self.gens() ] - """ - if len(p.variables()) > 1: - raise ValueError("not a univariate polynomial") - P = self.parent() - R = P.base_ring() - # Convert the coeficcients to the parent's base ring, - # because a priori they might live in an (unnecessarily) - # larger ring for which P.sum() would fail below. - cs = [ R(c) for c in p.coefficients(sparse=False) ] - return P.sum( cs[k]*(self**k) for k in range(len(cs)) ) + # Now we use basic linear algebra to find the coefficients, + # of the matrices-as-vectors-linear-combination, which should + # work for the original algebra basis too. + A = matrix(self.base_ring(), oper_vecs) + # We used the isometry on the left-hand side already, but we + # still need to do it for the right-hand side. Recall that we + # wanted something that summed to the identity matrix. + b = _mat2vec( matrix.identity(self.base_ring(), self.dimension()) ) - def characteristic_polynomial(self): - """ - Return the characteristic polynomial of this element. + # Now if there's an identity element in the algebra, this + # should work. We solve on the left to avoid having to + # transpose the matrix "A". + return self.from_vector(A.solve_left(b)) - SETUP:: - sage: from mjo.eja.eja_algebra import RealCartesianProductEJA + def peirce_decomposition(self, c): + """ + The Peirce decomposition of this algebra relative to the + idempotent ``c``. - EXAMPLES: + In the future, this can be extended to a complete system of + orthogonal idempotents. - The rank of `R^3` is three, and the minimal polynomial of - the identity element is `(t-1)` from which it follows that - the characteristic polynomial should be `(t-1)^3`:: + INPUT: - sage: J = RealCartesianProductEJA(3) - sage: J.one().characteristic_polynomial() - t^3 - 3*t^2 + 3*t - 1 + - ``c`` -- an idempotent of this algebra. - Likewise, the characteristic of the zero element in the - rank-three algebra `R^{n}` should be `t^{3}`:: + OUTPUT: - sage: J = RealCartesianProductEJA(3) - sage: J.zero().characteristic_polynomial() - t^3 + A triple (J0, J5, J1) containing two subalgebras and one subspace + of this algebra, - The characteristic polynomial of an element should evaluate - to zero on that element:: + - ``J0`` -- the algebra on the eigenspace of ``c.operator()`` + corresponding to the eigenvalue zero. - sage: set_random_seed() - sage: x = RealCartesianProductEJA(3).random_element() - sage: p = x.characteristic_polynomial() - sage: x.apply_univariate_polynomial(p) - 0 + - ``J5`` -- the eigenspace (NOT a subalgebra) of ``c.operator()`` + corresponding to the eigenvalue one-half. - """ - p = self.parent().characteristic_polynomial() - return p(*self.vector()) + - ``J1`` -- the algebra on the eigenspace of ``c.operator()`` + corresponding to the eigenvalue one. + These are the only possible eigenspaces for that operator, and this + algebra is a direct sum of them. The spaces ``J0`` and ``J1`` are + orthogonal, and are subalgebras of this algebra with the appropriate + restrictions. - def inner_product(self, other): - """ - Return the parent algebra's inner product of myself and ``other``. + SETUP:: - SETUP:: + sage: from mjo.eja.eja_algebra import random_eja, RealSymmetricEJA - sage: from mjo.eja.eja_algebra import ( - ....: ComplexHermitianEJA, - ....: JordanSpinEJA, - ....: QuaternionHermitianEJA, - ....: RealSymmetricEJA, - ....: random_eja) + EXAMPLES: - EXAMPLES: + The canonical example comes from the symmetric matrices, which + decompose into diagonal and off-diagonal parts:: + + sage: J = RealSymmetricEJA(3) + sage: C = matrix(QQ, [ [1,0,0], + ....: [0,1,0], + ....: [0,0,0] ]) + sage: c = J(C) + sage: J0,J5,J1 = J.peirce_decomposition(c) + sage: J0 + Euclidean Jordan algebra of dimension 1... + sage: J5 + Vector space of degree 6 and dimension 2... + sage: J1 + Euclidean Jordan algebra of dimension 3... + sage: J0.one().to_matrix() + [0 0 0] + [0 0 0] + [0 0 1] + sage: orig_df = AA.options.display_format + sage: AA.options.display_format = 'radical' + sage: J.from_vector(J5.basis()[0]).to_matrix() + [ 0 0 1/2*sqrt(2)] + [ 0 0 0] + [1/2*sqrt(2) 0 0] + sage: J.from_vector(J5.basis()[1]).to_matrix() + [ 0 0 0] + [ 0 0 1/2*sqrt(2)] + [ 0 1/2*sqrt(2) 0] + sage: AA.options.display_format = orig_df + sage: J1.one().to_matrix() + [1 0 0] + [0 1 0] + [0 0 0] - The inner product in the Jordan spin algebra is the usual - inner product on `R^n` (this example only works because the - basis for the Jordan algebra is the standard basis in `R^n`):: + TESTS: - sage: J = JordanSpinEJA(3) - sage: x = vector(QQ,[1,2,3]) - sage: y = vector(QQ,[4,5,6]) - sage: x.inner_product(y) - 32 - sage: J(x).inner_product(J(y)) - 32 + Every algebra decomposes trivially with respect to its identity + element:: - The inner product on `S^n` is ` = trace(X*Y)`, where - multiplication is the usual matrix multiplication in `S^n`, - so the inner product of the identity matrix with itself - should be the `n`:: - - sage: J = RealSymmetricEJA(3) - sage: J.one().inner_product(J.one()) - 3 - - Likewise, the inner product on `C^n` is ` = - Re(trace(X*Y))`, where we must necessarily take the real - part because the product of Hermitian matrices may not be - Hermitian:: - - sage: J = ComplexHermitianEJA(3) - sage: J.one().inner_product(J.one()) - 3 - - Ditto for the quaternions:: - - sage: J = QuaternionHermitianEJA(3) - sage: J.one().inner_product(J.one()) - 3 - - TESTS: - - Ensure that we can always compute an inner product, and that - it gives us back a real number:: - - sage: set_random_seed() - sage: J = random_eja() - sage: x = J.random_element() - sage: y = J.random_element() - sage: x.inner_product(y) in RR - True - - """ - P = self.parent() - if not other in P: - raise TypeError("'other' must live in the same algebra") - - return P.inner_product(self, other) - - - def operator_commutes_with(self, other): - """ - Return whether or not this element operator-commutes - with ``other``. - - SETUP:: - - sage: from mjo.eja.eja_algebra import random_eja - - EXAMPLES: - - The definition of a Jordan algebra says that any element - operator-commutes with its square:: - - sage: set_random_seed() - sage: x = random_eja().random_element() - sage: x.operator_commutes_with(x^2) - True - - TESTS: - - Test Lemma 1 from Chapter III of Koecher:: - - sage: set_random_seed() - sage: J = random_eja() - sage: u = J.random_element() - sage: v = J.random_element() - sage: lhs = u.operator_commutes_with(u*v) - sage: rhs = v.operator_commutes_with(u^2) - sage: lhs == rhs - True - - Test the first polarization identity from my notes, Koecher Chapter - III, or from Baes (2.3):: - - sage: set_random_seed() - sage: J = random_eja() - sage: x = J.random_element() - sage: y = J.random_element() - sage: Lx = x.operator() - sage: Ly = y.operator() - sage: Lxx = (x*x).operator() - sage: Lxy = (x*y).operator() - sage: bool(2*Lx*Lxy + Ly*Lxx == 2*Lxy*Lx + Lxx*Ly) - True - - Test the second polarization identity from my notes or from - Baes (2.4):: - - sage: set_random_seed() - sage: J = random_eja() - sage: x = J.random_element() - sage: y = J.random_element() - sage: z = J.random_element() - sage: Lx = x.operator() - sage: Ly = y.operator() - sage: Lz = z.operator() - sage: Lzy = (z*y).operator() - sage: Lxy = (x*y).operator() - sage: Lxz = (x*z).operator() - sage: bool(Lx*Lzy + Lz*Lxy + Ly*Lxz == Lzy*Lx + Lxy*Lz + Lxz*Ly) - True - - Test the third polarization identity from my notes or from - Baes (2.5):: - - sage: set_random_seed() - sage: J = random_eja() - sage: u = J.random_element() - sage: y = J.random_element() - sage: z = J.random_element() - sage: Lu = u.operator() - sage: Ly = y.operator() - sage: Lz = z.operator() - sage: Lzy = (z*y).operator() - sage: Luy = (u*y).operator() - sage: Luz = (u*z).operator() - sage: Luyz = (u*(y*z)).operator() - sage: lhs = Lu*Lzy + Lz*Luy + Ly*Luz - sage: rhs = Luyz + Ly*Lu*Lz + Lz*Lu*Ly - sage: bool(lhs == rhs) - True - - """ - if not other in self.parent(): - raise TypeError("'other' must live in the same algebra") - - A = self.operator() - B = other.operator() - return (A*B == B*A) - - - def det(self): - """ - Return my determinant, the product of my eigenvalues. - - SETUP:: - - sage: from mjo.eja.eja_algebra import (JordanSpinEJA, - ....: random_eja) - - EXAMPLES:: - - sage: J = JordanSpinEJA(2) - sage: e0,e1 = J.gens() - sage: x = sum( J.gens() ) - sage: x.det() - 0 - - :: - - sage: J = JordanSpinEJA(3) - sage: e0,e1,e2 = J.gens() - sage: x = sum( J.gens() ) - sage: x.det() - -1 - - TESTS: - - An element is invertible if and only if its determinant is - non-zero:: - - sage: set_random_seed() - sage: x = random_eja().random_element() - sage: x.is_invertible() == (x.det() != 0) - True - - """ - P = self.parent() - r = P.rank() - p = P._charpoly_coeff(0) - # The _charpoly_coeff function already adds the factor of - # -1 to ensure that _charpoly_coeff(0) is really what - # appears in front of t^{0} in the charpoly. However, - # we want (-1)^r times THAT for the determinant. - return ((-1)**r)*p(*self.vector()) - - - def inverse(self): - """ - Return the Jordan-multiplicative inverse of this element. - - ALGORITHM: + sage: set_random_seed() + sage: J = random_eja() + sage: J0,J5,J1 = J.peirce_decomposition(J.one()) + sage: J0.dimension() == 0 and J5.dimension() == 0 + True + sage: J1.superalgebra() == J and J1.dimension() == J.dimension() + True - We appeal to the quadratic representation as in Koecher's - Theorem 12 in Chapter III, Section 5. - - SETUP:: + The decomposition is into eigenspaces, and its components are + therefore necessarily orthogonal. Moreover, the identity + elements in the two subalgebras are the projections onto their + respective subspaces of the superalgebra's identity element:: - sage: from mjo.eja.eja_algebra import (JordanSpinEJA, - ....: random_eja) - - EXAMPLES: + sage: set_random_seed() + sage: J = random_eja() + sage: x = J.random_element() + sage: if not J.is_trivial(): + ....: while x.is_nilpotent(): + ....: x = J.random_element() + sage: c = x.subalgebra_idempotent() + sage: J0,J5,J1 = J.peirce_decomposition(c) + sage: ipsum = 0 + sage: for (w,y,z) in zip(J0.basis(), J5.basis(), J1.basis()): + ....: w = w.superalgebra_element() + ....: y = J.from_vector(y) + ....: z = z.superalgebra_element() + ....: ipsum += w.inner_product(y).abs() + ....: ipsum += w.inner_product(z).abs() + ....: ipsum += y.inner_product(z).abs() + sage: ipsum + 0 + sage: J1(c) == J1.one() + True + sage: J0(J.one() - c) == J0.one() + True - The inverse in the spin factor algebra is given in Alizadeh's - Example 11.11:: + """ + if not c.is_idempotent(): + raise ValueError("element is not idempotent: %s" % c) + + # Default these to what they should be if they turn out to be + # trivial, because eigenspaces_left() won't return eigenvalues + # corresponding to trivial spaces (e.g. it returns only the + # eigenspace corresponding to lambda=1 if you take the + # decomposition relative to the identity element). + trivial = self.subalgebra(()) + J0 = trivial # eigenvalue zero + J5 = VectorSpace(self.base_ring(), 0) # eigenvalue one-half + J1 = trivial # eigenvalue one + + for (eigval, eigspace) in c.operator().matrix().right_eigenspaces(): + if eigval == ~(self.base_ring()(2)): + J5 = eigspace + else: + gens = tuple( self.from_vector(b) for b in eigspace.basis() ) + subalg = self.subalgebra(gens, check_axioms=False) + if eigval == 0: + J0 = subalg + elif eigval == 1: + J1 = subalg + else: + raise ValueError("unexpected eigenvalue: %s" % eigval) - sage: set_random_seed() - sage: n = ZZ.random_element(1,10) - sage: J = JordanSpinEJA(n) - sage: x = J.random_element() - sage: while not x.is_invertible(): - ....: x = J.random_element() - sage: x_vec = x.vector() - sage: x0 = x_vec[0] - sage: x_bar = x_vec[1:] - sage: coeff = ~(x0^2 - x_bar.inner_product(x_bar)) - sage: inv_vec = x_vec.parent()([x0] + (-x_bar).list()) - sage: x_inverse = coeff*inv_vec - sage: x.inverse() == J(x_inverse) - True + return (J0, J5, J1) - TESTS: - The identity element is its own inverse:: + def random_element(self, thorough=False): + r""" + Return a random element of this algebra. - sage: set_random_seed() - sage: J = random_eja() - sage: J.one().inverse() == J.one() - True + Our algebra superclass method only returns a linear + combination of at most two basis elements. We instead + want the vector space "random element" method that + returns a more diverse selection. - If an element has an inverse, it acts like one:: + INPUT: - sage: set_random_seed() - sage: J = random_eja() - sage: x = J.random_element() - sage: (not x.is_invertible()) or (x.inverse()*x == J.one()) - True + - ``thorough`` -- (boolean; default False) whether or not we + should generate irrational coefficients for the random + element when our base ring is irrational; this slows the + algebra operations to a crawl, but any truly random method + should include them - The inverse of the inverse is what we started with:: + """ + # For a general base ring... maybe we can trust this to do the + # right thing? Unlikely, but. + V = self.vector_space() + v = V.random_element() - sage: set_random_seed() - sage: J = random_eja() - sage: x = J.random_element() - sage: (not x.is_invertible()) or (x.inverse().inverse() == x) - True + if self.base_ring() is AA: + # The "random element" method of the algebraic reals is + # stupid at the moment, and only returns integers between + # -2 and 2, inclusive: + # + # https://trac.sagemath.org/ticket/30875 + # + # Instead, we implement our own "random vector" method, + # and then coerce that into the algebra. We use the vector + # space degree here instead of the dimension because a + # subalgebra could (for example) be spanned by only two + # vectors, each with five coordinates. We need to + # generate all five coordinates. + if thorough: + v *= QQbar.random_element().real() + else: + v *= QQ.random_element() - The zero element is never invertible:: + return self.from_vector(V.coordinate_vector(v)) - sage: set_random_seed() - sage: J = random_eja().zero().inverse() - Traceback (most recent call last): - ... - ValueError: element is not invertible + def random_elements(self, count, thorough=False): + """ + Return ``count`` random elements as a tuple. - """ - if not self.is_invertible(): - raise ValueError("element is not invertible") + INPUT: - return (~self.quadratic_representation())(self) + - ``thorough`` -- (boolean; default False) whether or not we + should generate irrational coefficients for the random + elements when our base ring is irrational; this slows the + algebra operations to a crawl, but any truly random method + should include them + SETUP:: - def is_invertible(self): - """ - Return whether or not this element is invertible. + sage: from mjo.eja.eja_algebra import JordanSpinEJA - We can't use the superclass method because it relies on - the algebra being associative. + EXAMPLES:: - ALGORITHM: + sage: J = JordanSpinEJA(3) + sage: x,y,z = J.random_elements(3) + sage: all( [ x in J, y in J, z in J ]) + True + sage: len( J.random_elements(10) ) == 10 + True - The usual way to do this is to check if the determinant is - zero, but we need the characteristic polynomial for the - determinant. The minimal polynomial is a lot easier to get, - so we use Corollary 2 in Chapter V of Koecher to check - whether or not the paren't algebra's zero element is a root - of this element's minimal polynomial. + """ + return tuple( self.random_element(thorough) + for idx in range(count) ) - SETUP:: - sage: from mjo.eja.eja_algebra import random_eja + @cached_method + def _charpoly_coefficients(self): + r""" + The `r` polynomial coefficients of the "characteristic polynomial + of" function. - TESTS: + SETUP:: - The identity element is always invertible:: + sage: from mjo.eja.eja_algebra import random_eja - sage: set_random_seed() - sage: J = random_eja() - sage: J.one().is_invertible() - True + TESTS: - The zero element is never invertible:: + The theory shows that these are all homogeneous polynomials of + a known degree:: - sage: set_random_seed() - sage: J = random_eja() - sage: J.zero().is_invertible() - False + sage: set_random_seed() + sage: J = random_eja() + sage: all(p.is_homogeneous() for p in J._charpoly_coefficients()) + True - """ - zero = self.parent().zero() - p = self.minimal_polynomial() - return not (p(zero) == zero) + """ + n = self.dimension() + R = self.coordinate_polynomial_ring() + vars = R.gens() + F = R.fraction_field() + + def L_x_i_j(i,j): + # From a result in my book, these are the entries of the + # basis representation of L_x. + return sum( vars[k]*self.monomial(k).operator().matrix()[i,j] + for k in range(n) ) + + L_x = matrix(F, n, n, L_x_i_j) + + r = None + if self.rank.is_in_cache(): + r = self.rank() + # There's no need to pad the system with redundant + # columns if we *know* they'll be redundant. + n = r + + # Compute an extra power in case the rank is equal to + # the dimension (otherwise, we would stop at x^(r-1)). + x_powers = [ (L_x**k)*self.one().to_vector() + for k in range(n+1) ] + A = matrix.column(F, x_powers[:n]) + AE = A.extended_echelon_form() + E = AE[:,n:] + A_rref = AE[:,:n] + if r is None: + r = A_rref.rank() + b = x_powers[r] + + # The theory says that only the first "r" coefficients are + # nonzero, and they actually live in the original polynomial + # ring and not the fraction field. We negate them because in + # the actual characteristic polynomial, they get moved to the + # other side where x^r lives. We don't bother to trim A_rref + # down to a square matrix and solve the resulting system, + # because the upper-left r-by-r portion of A_rref is + # guaranteed to be the identity matrix, so e.g. + # + # A_rref.solve_right(Y) + # + # would just be returning Y. + return (-E*b)[:r].change_ring(R) + @cached_method + def rank(self): + r""" + Return the rank of this EJA. - def is_nilpotent(self): - """ - Return whether or not some power of this element is zero. + This is a cached method because we know the rank a priori for + all of the algebras we can construct. Thus we can avoid the + expensive ``_charpoly_coefficients()`` call unless we truly + need to compute the whole characteristic polynomial. - The superclass method won't work unless we're in an - associative algebra, and we aren't. However, we generate - an assocoative subalgebra and we're nilpotent there if and - only if we're nilpotent here (probably). + SETUP:: - SETUP:: + sage: from mjo.eja.eja_algebra import (HadamardEJA, + ....: JordanSpinEJA, + ....: RealSymmetricEJA, + ....: ComplexHermitianEJA, + ....: QuaternionHermitianEJA, + ....: random_eja) - sage: from mjo.eja.eja_algebra import random_eja + EXAMPLES: - TESTS: + The rank of the Jordan spin algebra is always two:: - The identity element is never nilpotent:: + sage: JordanSpinEJA(2).rank() + 2 + sage: JordanSpinEJA(3).rank() + 2 + sage: JordanSpinEJA(4).rank() + 2 - sage: set_random_seed() - sage: random_eja().one().is_nilpotent() - False + The rank of the `n`-by-`n` Hermitian real, complex, or + quaternion matrices is `n`:: - The additive identity is always nilpotent:: + sage: RealSymmetricEJA(4).rank() + 4 + sage: ComplexHermitianEJA(3).rank() + 3 + sage: QuaternionHermitianEJA(2).rank() + 2 - sage: set_random_seed() - sage: random_eja().zero().is_nilpotent() - True + TESTS: - """ - # The element we're going to call "is_nilpotent()" on. - # Either myself, interpreted as an element of a finite- - # dimensional algebra, or an element of an associative - # subalgebra. - elt = None + Ensure that every EJA that we know how to construct has a + positive integer rank, unless the algebra is trivial in + which case its rank will be zero:: - if self.parent().is_associative(): - elt = FiniteDimensionalAlgebraElement(self.parent(), self) - else: - V = self.span_of_powers() - assoc_subalg = self.subalgebra_generated_by() - # Mis-design warning: the basis used for span_of_powers() - # and subalgebra_generated_by() must be the same, and in - # the same order! - elt = assoc_subalg(V.coordinates(self.vector())) + sage: set_random_seed() + sage: J = random_eja() + sage: r = J.rank() + sage: r in ZZ + True + sage: r > 0 or (r == 0 and J.is_trivial()) + True - # Recursive call, but should work since elt lives in an - # associative algebra. - return elt.is_nilpotent() + Ensure that computing the rank actually works, since the ranks + of all simple algebras are known and will be cached by default:: + sage: set_random_seed() # long time + sage: J = random_eja() # long time + sage: cached = J.rank() # long time + sage: J.rank.clear_cache() # long time + sage: J.rank() == cached # long time + True - def is_regular(self): - """ - Return whether or not this is a regular element. + """ + return len(self._charpoly_coefficients()) - SETUP:: - sage: from mjo.eja.eja_algebra import JordanSpinEJA + def subalgebra(self, basis, **kwargs): + r""" + Create a subalgebra of this algebra from the given basis. + """ + from mjo.eja.eja_subalgebra import FiniteDimensionalEJASubalgebra + return FiniteDimensionalEJASubalgebra(self, basis, **kwargs) - EXAMPLES: - The identity element always has degree one, but any element - linearly-independent from it is regular:: + def vector_space(self): + """ + Return the vector space that underlies this algebra. - sage: J = JordanSpinEJA(5) - sage: J.one().is_regular() - False - sage: e0, e1, e2, e3, e4 = J.gens() # e0 is the identity - sage: for x in J.gens(): - ....: (J.one() + x).is_regular() - False - True - True - True - True - - """ - return self.degree() == self.parent().rank() + SETUP:: + sage: from mjo.eja.eja_algebra import RealSymmetricEJA - def degree(self): - """ - Compute the degree of this element the straightforward way - according to the definition; by appending powers to a list - and figuring out its dimension (that is, whether or not - they're linearly dependent). - - SETUP:: + EXAMPLES:: - sage: from mjo.eja.eja_algebra import JordanSpinEJA - - EXAMPLES:: - - sage: J = JordanSpinEJA(4) - sage: J.one().degree() - 1 - sage: e0,e1,e2,e3 = J.gens() - sage: (e0 - e1).degree() - 2 + sage: J = RealSymmetricEJA(2) + sage: J.vector_space() + Vector space of dimension 3 over... - In the spin factor algebra (of rank two), all elements that - aren't multiples of the identity are regular:: + """ + return self.zero().to_vector().parent().ambient_vector_space() - sage: set_random_seed() - sage: n = ZZ.random_element(1,10) - sage: J = JordanSpinEJA(n) - sage: x = J.random_element() - sage: x == x.coefficient(0)*J.one() or x.degree() == 2 - True - """ - return self.span_of_powers().dimension() +class RationalBasisEJA(FiniteDimensionalEJA): + r""" + Algebras whose supplied basis elements have all rational entries. - def left_matrix(self): - """ - Our parent class defines ``left_matrix`` and ``matrix`` - methods whose names are misleading. We don't want them. - """ - raise NotImplementedError("use operator().matrix() instead") + SETUP:: - matrix = left_matrix - - - def minimal_polynomial(self): - """ - Return the minimal polynomial of this element, - as a function of the variable `t`. + sage: from mjo.eja.eja_algebra import BilinearFormEJA - ALGORITHM: - - We restrict ourselves to the associative subalgebra - generated by this element, and then return the minimal - polynomial of this element's operator matrix (in that - subalgebra). This works by Baes Proposition 2.3.16. - - SETUP:: - - sage: from mjo.eja.eja_algebra import (JordanSpinEJA, - ....: random_eja) - - TESTS: - - The minimal polynomial of the identity and zero elements are - always the same:: - - sage: set_random_seed() - sage: J = random_eja() - sage: J.one().minimal_polynomial() - t - 1 - sage: J.zero().minimal_polynomial() - t - - The degree of an element is (by one definition) the degree - of its minimal polynomial:: - - sage: set_random_seed() - sage: x = random_eja().random_element() - sage: x.degree() == x.minimal_polynomial().degree() - True - - The minimal polynomial and the characteristic polynomial coincide - and are known (see Alizadeh, Example 11.11) for all elements of - the spin factor algebra that aren't scalar multiples of the - identity:: - - sage: set_random_seed() - sage: n = ZZ.random_element(2,10) - sage: J = JordanSpinEJA(n) - sage: y = J.random_element() - sage: while y == y.coefficient(0)*J.one(): - ....: y = J.random_element() - sage: y0 = y.vector()[0] - sage: y_bar = y.vector()[1:] - sage: actual = y.minimal_polynomial() - sage: t = PolynomialRing(J.base_ring(),'t').gen(0) - sage: expected = t^2 - 2*y0*t + (y0^2 - norm(y_bar)^2) - sage: bool(actual == expected) - True - - The minimal polynomial should always kill its element:: - - sage: set_random_seed() - sage: x = random_eja().random_element() - sage: p = x.minimal_polynomial() - sage: x.apply_univariate_polynomial(p) - 0 - - """ - V = self.span_of_powers() - assoc_subalg = self.subalgebra_generated_by() - # Mis-design warning: the basis used for span_of_powers() - # and subalgebra_generated_by() must be the same, and in - # the same order! - elt = assoc_subalg(V.coordinates(self.vector())) - return elt.operator().minimal_polynomial() - - - - def natural_representation(self): - """ - Return a more-natural representation of this element. - - Every finite-dimensional Euclidean Jordan Algebra is a - direct sum of five simple algebras, four of which comprise - Hermitian matrices. This method returns the original - "natural" representation of this element as a Hermitian - matrix, if it has one. If not, you get the usual representation. - - SETUP:: - - sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA, - ....: QuaternionHermitianEJA) - - EXAMPLES:: - - sage: J = ComplexHermitianEJA(3) - sage: J.one() - e0 + e5 + e8 - sage: J.one().natural_representation() - [1 0 0 0 0 0] - [0 1 0 0 0 0] - [0 0 1 0 0 0] - [0 0 0 1 0 0] - [0 0 0 0 1 0] - [0 0 0 0 0 1] - - :: - - sage: J = QuaternionHermitianEJA(3) - sage: J.one() - e0 + e9 + e14 - sage: J.one().natural_representation() - [1 0 0 0 0 0 0 0 0 0 0 0] - [0 1 0 0 0 0 0 0 0 0 0 0] - [0 0 1 0 0 0 0 0 0 0 0 0] - [0 0 0 1 0 0 0 0 0 0 0 0] - [0 0 0 0 1 0 0 0 0 0 0 0] - [0 0 0 0 0 1 0 0 0 0 0 0] - [0 0 0 0 0 0 1 0 0 0 0 0] - [0 0 0 0 0 0 0 1 0 0 0 0] - [0 0 0 0 0 0 0 0 1 0 0 0] - [0 0 0 0 0 0 0 0 0 1 0 0] - [0 0 0 0 0 0 0 0 0 0 1 0] - [0 0 0 0 0 0 0 0 0 0 0 1] - - """ - B = self.parent().natural_basis() - W = B[0].matrix_space() - return W.linear_combination(zip(self.vector(), B)) - - - def operator(self): - """ - Return the left-multiplication-by-this-element - operator on the ambient algebra. - - SETUP:: - - sage: from mjo.eja.eja_algebra import random_eja - - TESTS:: - - sage: set_random_seed() - sage: J = random_eja() - sage: x = J.random_element() - sage: y = J.random_element() - sage: x.operator()(y) == x*y - True - sage: y.operator()(x) == x*y - True - - """ - P = self.parent() - fda_elt = FiniteDimensionalAlgebraElement(P, self) - return FiniteDimensionalEuclideanJordanAlgebraOperator( - P, - P, - fda_elt.matrix().transpose() ) - - - def quadratic_representation(self, other=None): - """ - Return the quadratic representation of this element. - - SETUP:: - - sage: from mjo.eja.eja_algebra import (JordanSpinEJA, - ....: random_eja) - - EXAMPLES: - - The explicit form in the spin factor algebra is given by - Alizadeh's Example 11.12:: - - sage: set_random_seed() - sage: n = ZZ.random_element(1,10) - sage: J = JordanSpinEJA(n) - sage: x = J.random_element() - sage: x_vec = x.vector() - sage: x0 = x_vec[0] - sage: x_bar = x_vec[1:] - sage: A = matrix(QQ, 1, [x_vec.inner_product(x_vec)]) - sage: B = 2*x0*x_bar.row() - sage: C = 2*x0*x_bar.column() - sage: D = matrix.identity(QQ, n-1) - sage: D = (x0^2 - x_bar.inner_product(x_bar))*D - sage: D = D + 2*x_bar.tensor_product(x_bar) - sage: Q = matrix.block(2,2,[A,B,C,D]) - sage: Q == x.quadratic_representation().matrix() - True - - Test all of the properties from Theorem 11.2 in Alizadeh:: - - sage: set_random_seed() - sage: J = random_eja() - sage: x = J.random_element() - sage: y = J.random_element() - sage: Lx = x.operator() - sage: Lxx = (x*x).operator() - sage: Qx = x.quadratic_representation() - sage: Qy = y.quadratic_representation() - sage: Qxy = x.quadratic_representation(y) - sage: Qex = J.one().quadratic_representation(x) - sage: n = ZZ.random_element(10) - sage: Qxn = (x^n).quadratic_representation() - - Property 1: - - sage: 2*Qxy == (x+y).quadratic_representation() - Qx - Qy - True - - Property 2 (multiply on the right for :trac:`28272`): - - sage: alpha = QQ.random_element() - sage: (alpha*x).quadratic_representation() == Qx*(alpha^2) - True - - Property 3: - - sage: not x.is_invertible() or ( Qx(x.inverse()) == x ) - True + EXAMPLES: - sage: not x.is_invertible() or ( - ....: ~Qx - ....: == - ....: x.inverse().quadratic_representation() ) - True + The supplied basis is orthonormalized by default:: - sage: Qxy(J.one()) == x*y - True + sage: B = matrix(QQ, [[1, 0, 0], [0, 25, -32], [0, -32, 41]]) + sage: J = BilinearFormEJA(B) + sage: J.matrix_basis() + ( + [1] [ 0] [ 0] + [0] [1/5] [32/5] + [0], [ 0], [ 5] + ) - Property 4: - - sage: not x.is_invertible() or ( - ....: x.quadratic_representation(x.inverse())*Qx - ....: == Qx*x.quadratic_representation(x.inverse()) ) - True + """ + def __init__(self, + basis, + jordan_product, + inner_product, + field=AA, + check_field=True, + **kwargs): + + if check_field: + # Abuse the check_field parameter to check that the entries of + # out basis (in ambient coordinates) are in the field QQ. + # Use _all2list to get the vector coordinates of octonion + # entries and not the octonions themselves (which are not + # rational). + if not all( all(b_i in QQ for b_i in _all2list(b)) + for b in basis ): + raise TypeError("basis not rational") + + super().__init__(basis, + jordan_product, + inner_product, + field=field, + check_field=check_field, + **kwargs) + + self._rational_algebra = None + if field is not QQ: + # There's no point in constructing the extra algebra if this + # one is already rational. + # + # Note: the same Jordan and inner-products work here, + # because they are necessarily defined with respect to + # ambient coordinates and not any particular basis. + self._rational_algebra = FiniteDimensionalEJA( + basis, + jordan_product, + inner_product, + field=QQ, + associative=self.is_associative(), + orthonormalize=False, + check_field=False, + check_axioms=False) - sage: not x.is_invertible() or ( - ....: x.quadratic_representation(x.inverse())*Qx - ....: == - ....: 2*x.operator()*Qex - Qx ) - True - - sage: 2*x.operator()*Qex - Qx == Lxx - True - - Property 5: - - sage: Qy(x).quadratic_representation() == Qy*Qx*Qy - True - - Property 6: + @cached_method + def _charpoly_coefficients(self): + r""" + SETUP:: - sage: Qxn == (Qx)^n - True + sage: from mjo.eja.eja_algebra import (BilinearFormEJA, + ....: JordanSpinEJA) - Property 7: + EXAMPLES: - sage: not x.is_invertible() or ( - ....: Qx*x.inverse().operator() == Lx ) - True + The base ring of the resulting polynomial coefficients is what + it should be, and not the rationals (unless the algebra was + already over the rationals):: - Property 8: - - sage: not x.operator_commutes_with(y) or ( - ....: Qx(y)^n == Qxn(y^n) ) - True - - """ - if other is None: - other=self - elif not other in self.parent(): - raise TypeError("'other' must live in the same algebra") - - L = self.operator() - M = other.operator() - return ( L*M + M*L - (self*other).operator() ) - - - def span_of_powers(self): - """ - Return the vector space spanned by successive powers of - this element. - """ - # The dimension of the subalgebra can't be greater than - # the big algebra, so just put everything into a list - # and let span() get rid of the excess. - # - # We do the extra ambient_vector_space() in case we're messing - # with polynomials and the direct parent is a module. - V = self.parent().vector_space() - return V.span( (self**d).vector() for d in xrange(V.dimension()) ) - - - def subalgebra_generated_by(self): - """ - Return the associative subalgebra of the parent EJA generated - by this element. - - SETUP:: - - sage: from mjo.eja.eja_algebra import random_eja - - TESTS:: - - sage: set_random_seed() - sage: x = random_eja().random_element() - sage: x.subalgebra_generated_by().is_associative() - True - - Squaring in the subalgebra should work the same as in - the superalgebra:: - - sage: set_random_seed() - sage: x = random_eja().random_element() - sage: u = x.subalgebra_generated_by().random_element() - sage: u.operator()(u) == u^2 - True - - """ - # First get the subspace spanned by the powers of myself... - V = self.span_of_powers() - F = self.base_ring() - - # Now figure out the entries of the right-multiplication - # matrix for the successive basis elements b0, b1,... of - # that subspace. - mats = [] - for b_right in V.basis(): - eja_b_right = self.parent()(b_right) - b_right_rows = [] - # The first row of the right-multiplication matrix by - # b1 is what we get if we apply that matrix to b1. The - # second row of the right multiplication matrix by b1 - # is what we get when we apply that matrix to b2... - # - # IMPORTANT: this assumes that all vectors are COLUMN - # vectors, unlike our superclass (which uses row vectors). - for b_left in V.basis(): - eja_b_left = self.parent()(b_left) - # Multiply in the original EJA, but then get the - # coordinates from the subalgebra in terms of its - # basis. - this_row = V.coordinates((eja_b_left*eja_b_right).vector()) - b_right_rows.append(this_row) - b_right_matrix = matrix(F, b_right_rows) - mats.append(b_right_matrix) - - # It's an algebra of polynomials in one element, and EJAs - # are power-associative. - # - # TODO: choose generator names intelligently. - return FiniteDimensionalEuclideanJordanAlgebra(F, mats, assume_associative=True, names='f') - - - def subalgebra_idempotent(self): - """ - Find an idempotent in the associative subalgebra I generate - using Proposition 2.3.5 in Baes. - - SETUP:: - - sage: from mjo.eja.eja_algebra import random_eja - - TESTS:: - - sage: set_random_seed() - sage: J = random_eja() - sage: x = J.random_element() - sage: while x.is_nilpotent(): - ....: x = J.random_element() - sage: c = x.subalgebra_idempotent() - sage: c^2 == c - True - - """ - if self.is_nilpotent(): - raise ValueError("this only works with non-nilpotent elements!") - - V = self.span_of_powers() - J = self.subalgebra_generated_by() - # Mis-design warning: the basis used for span_of_powers() - # and subalgebra_generated_by() must be the same, and in - # the same order! - u = J(V.coordinates(self.vector())) - - # The image of the matrix of left-u^m-multiplication - # will be minimal for some natural number s... - s = 0 - minimal_dim = V.dimension() - for i in xrange(1, V.dimension()): - this_dim = (u**i).operator().matrix().image().dimension() - if this_dim < minimal_dim: - minimal_dim = this_dim - s = i - - # Now minimal_matrix should correspond to the smallest - # non-zero subspace in Baes's (or really, Koecher's) - # proposition. - # - # However, we need to restrict the matrix to work on the - # subspace... or do we? Can't we just solve, knowing that - # A(c) = u^(s+1) should have a solution in the big space, - # too? - # - # Beware, solve_right() means that we're using COLUMN vectors. - # Our FiniteDimensionalAlgebraElement superclass uses rows. - u_next = u**(s+1) - A = u_next.operator().matrix() - c_coordinates = A.solve_right(u_next.vector()) - - # Now c_coordinates is the idempotent we want, but it's in - # the coordinate system of the subalgebra. - # - # We need the basis for J, but as elements of the parent algebra. - # - basis = [self.parent(v) for v in V.basis()] - return self.parent().linear_combination(zip(c_coordinates, basis)) + sage: J = JordanSpinEJA(3) + sage: J._charpoly_coefficients() + (X1^2 - X2^2 - X3^2, -2*X1) + sage: a0 = J._charpoly_coefficients()[0] + sage: J.base_ring() + Algebraic Real Field + sage: a0.base_ring() + Algebraic Real Field + """ + if self._rational_algebra is None: + # There's no need to construct *another* algebra over the + # rationals if this one is already over the + # rationals. Likewise, if we never orthonormalized our + # basis, we might as well just use the given one. + return super()._charpoly_coefficients() + + # Do the computation over the rationals. The answer will be + # the same, because all we've done is a change of basis. + # Then, change back from QQ to our real base ring + a = ( a_i.change_ring(self.base_ring()) + for a_i in self._rational_algebra._charpoly_coefficients() ) + + if self._deortho_matrix is None: + # This can happen if our base ring was, say, AA and we + # chose not to (or didn't need to) orthonormalize. It's + # still faster to do the computations over QQ even if + # the numbers in the boxes stay the same. + return tuple(a) + + # Otherwise, convert the coordinate variables back to the + # deorthonormalized ones. + R = self.coordinate_polynomial_ring() + from sage.modules.free_module_element import vector + X = vector(R, R.gens()) + BX = self._deortho_matrix*X + + subs_dict = { X[i]: BX[i] for i in range(len(X)) } + return tuple( a_i.subs(subs_dict) for a_i in a ) + +class ConcreteEJA(FiniteDimensionalEJA): + r""" + A class for the Euclidean Jordan algebras that we know by name. + + These are the Jordan algebras whose basis, multiplication table, + rank, and so on are known a priori. More to the point, they are + the Euclidean Jordan algebras for which we are able to conjure up + a "random instance." - def trace(self): - """ - Return my trace, the sum of my eigenvalues. + SETUP:: - SETUP:: + sage: from mjo.eja.eja_algebra import ConcreteEJA - sage: from mjo.eja.eja_algebra import (JordanSpinEJA, - ....: RealCartesianProductEJA, - ....: random_eja) + TESTS: - EXAMPLES:: + Our basis is normalized with respect to the algebra's inner + product, unless we specify otherwise:: - sage: J = JordanSpinEJA(3) - sage: x = sum(J.gens()) - sage: x.trace() - 2 + sage: set_random_seed() + sage: J = ConcreteEJA.random_instance() + sage: all( b.norm() == 1 for b in J.gens() ) + True - :: + Since our basis is orthonormal with respect to the algebra's inner + product, and since we know that this algebra is an EJA, any + left-multiplication operator's matrix will be symmetric because + natural->EJA basis representation is an isometry and within the + EJA the operator is self-adjoint by the Jordan axiom:: - sage: J = RealCartesianProductEJA(5) - sage: J.one().trace() - 5 + sage: set_random_seed() + sage: J = ConcreteEJA.random_instance() + sage: x = J.random_element() + sage: x.operator().is_self_adjoint() + True + """ - TESTS: + @staticmethod + def _max_random_instance_size(): + """ + Return an integer "size" that is an upper bound on the size of + this algebra when it is used in a random test + case. Unfortunately, the term "size" is ambiguous -- when + dealing with `R^n` under either the Hadamard or Jordan spin + product, the "size" refers to the dimension `n`. When dealing + with a matrix algebra (real symmetric or complex/quaternion + Hermitian), it refers to the size of the matrix, which is far + less than the dimension of the underlying vector space. + + This method must be implemented in each subclass. + """ + raise NotImplementedError - The trace of an element is a real number:: + @classmethod + def random_instance(cls, *args, **kwargs): + """ + Return a random instance of this type of algebra. - sage: set_random_seed() - sage: J = random_eja() - sage: J.random_element().trace() in J.base_ring() - True + This method should be implemented in each subclass. + """ + from sage.misc.prandom import choice + eja_class = choice(cls.__subclasses__()) - """ - P = self.parent() - r = P.rank() - p = P._charpoly_coeff(r-1) - # The _charpoly_coeff function already adds the factor of - # -1 to ensure that _charpoly_coeff(r-1) is really what - # appears in front of t^{r-1} in the charpoly. However, - # we want the negative of THAT for the trace. - return -p(*self.vector()) + # These all bubble up to the RationalBasisEJA superclass + # constructor, so any (kw)args valid there are also valid + # here. + return eja_class.random_instance(*args, **kwargs) - def trace_inner_product(self, other): - """ - Return the trace inner product of myself and ``other``. +class MatrixEJA: + @staticmethod + def _denormalized_basis(A): + """ + Returns a basis for the space of complex Hermitian n-by-n matrices. - SETUP:: + Why do we embed these? Basically, because all of numerical linear + algebra assumes that you're working with vectors consisting of `n` + entries from a field and scalars from the same field. There's no way + to tell SageMath that (for example) the vectors contain complex + numbers, while the scalar field is real. - sage: from mjo.eja.eja_algebra import random_eja + SETUP:: - TESTS: + sage: from mjo.hurwitz import (ComplexMatrixAlgebra, + ....: QuaternionMatrixAlgebra, + ....: OctonionMatrixAlgebra) + sage: from mjo.eja.eja_algebra import MatrixEJA - The trace inner product is commutative:: + TESTS:: - sage: set_random_seed() - sage: J = random_eja() - sage: x = J.random_element(); y = J.random_element() - sage: x.trace_inner_product(y) == y.trace_inner_product(x) - True + sage: set_random_seed() + sage: n = ZZ.random_element(1,5) + sage: A = MatrixSpace(QQ, n) + sage: B = MatrixEJA._denormalized_basis(A) + sage: all( M.is_hermitian() for M in B) + True - The trace inner product is bilinear:: + :: - sage: set_random_seed() - sage: J = random_eja() - sage: x = J.random_element() - sage: y = J.random_element() - sage: z = J.random_element() - sage: a = QQ.random_element(); - sage: actual = (a*(x+z)).trace_inner_product(y) - sage: expected = ( a*x.trace_inner_product(y) + - ....: a*z.trace_inner_product(y) ) - sage: actual == expected - True - sage: actual = x.trace_inner_product(a*(y+z)) - sage: expected = ( a*x.trace_inner_product(y) + - ....: a*x.trace_inner_product(z) ) - sage: actual == expected - True + sage: set_random_seed() + sage: n = ZZ.random_element(1,5) + sage: A = ComplexMatrixAlgebra(n, scalars=QQ) + sage: B = MatrixEJA._denormalized_basis(A) + sage: all( M.is_hermitian() for M in B) + True - The trace inner product satisfies the compatibility - condition in the definition of a Euclidean Jordan algebra:: + :: - sage: set_random_seed() - sage: J = random_eja() - sage: x = J.random_element() - sage: y = J.random_element() - sage: z = J.random_element() - sage: (x*y).trace_inner_product(z) == y.trace_inner_product(x*z) - True + sage: set_random_seed() + sage: n = ZZ.random_element(1,5) + sage: A = QuaternionMatrixAlgebra(n, scalars=QQ) + sage: B = MatrixEJA._denormalized_basis(A) + sage: all( M.is_hermitian() for M in B ) + True - """ - if not other in self.parent(): - raise TypeError("'other' must live in the same algebra") + :: - return (self*other).trace() + sage: set_random_seed() + sage: n = ZZ.random_element(1,5) + sage: A = OctonionMatrixAlgebra(n, scalars=QQ) + sage: B = MatrixEJA._denormalized_basis(A) + sage: all( M.is_hermitian() for M in B ) + True + """ + # These work for real MatrixSpace, whose monomials only have + # two coordinates (because the last one would always be "1"). + es = A.base_ring().gens() + gen = lambda A,m: A.monomial(m[:2]) + + if hasattr(A, 'entry_algebra_gens'): + # We've got a MatrixAlgebra, and its monomials will have + # three coordinates. + es = A.entry_algebra_gens() + gen = lambda A,m: A.monomial(m) + + basis = [] + for i in range(A.nrows()): + for j in range(i+1): + if i == j: + E_ii = gen(A, (i,j,es[0])) + basis.append(E_ii) + else: + for e in es: + E_ij = gen(A, (i,j,e)) + E_ij += E_ij.conjugate_transpose() + basis.append(E_ij) + + return tuple( basis ) -class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra): - """ - Return the Euclidean Jordan Algebra corresponding to the set - `R^n` under the Hadamard product. + @staticmethod + def jordan_product(X,Y): + return (X*Y + Y*X)/2 - Note: this is nothing more than the Cartesian product of ``n`` - copies of the spin algebra. Once Cartesian product algebras - are implemented, this can go. + @staticmethod + def trace_inner_product(X,Y): + r""" + A trace inner-product for matrices that aren't embedded in the + reals. It takes MATRICES as arguments, not EJA elements. - SETUP:: + SETUP:: - sage: from mjo.eja.eja_algebra import RealCartesianProductEJA + sage: from mjo.eja.eja_algebra import (RealSymmetricEJA, + ....: ComplexHermitianEJA, + ....: QuaternionHermitianEJA, + ....: OctonionHermitianEJA) - EXAMPLES: + EXAMPLES:: - This multiplication table can be verified by hand:: + sage: J = RealSymmetricEJA(2,field=QQ,orthonormalize=False) + sage: I = J.one().to_matrix() + sage: J.trace_inner_product(I, -I) + -2 - sage: J = RealCartesianProductEJA(3) - sage: e0,e1,e2 = J.gens() - sage: e0*e0 - e0 - sage: e0*e1 - 0 - sage: e0*e2 - 0 - sage: e1*e1 - e1 - sage: e1*e2 - 0 - sage: e2*e2 - e2 + :: - """ - @staticmethod - def __classcall_private__(cls, n, field=QQ): - # The FiniteDimensionalAlgebra constructor takes a list of - # matrices, the ith representing right multiplication by the ith - # basis element in the vector space. So if e_1 = (1,0,0), then - # right (Hadamard) multiplication of x by e_1 picks out the first - # component of x; and likewise for the ith basis element e_i. - Qs = [ matrix(field, n, n, lambda k,j: 1*(k == j == i)) - for i in xrange(n) ] - - fdeja = super(RealCartesianProductEJA, cls) - return fdeja.__classcall_private__(cls, field, Qs, rank=n) + sage: J = ComplexHermitianEJA(2,field=QQ,orthonormalize=False) + sage: I = J.one().to_matrix() + sage: J.trace_inner_product(I, -I) + -2 - def inner_product(self, x, y): - return _usual_ip(x,y) + :: + sage: J = QuaternionHermitianEJA(2,field=QQ,orthonormalize=False) + sage: I = J.one().to_matrix() + sage: J.trace_inner_product(I, -I) + -2 -def random_eja(): - """ - Return a "random" finite-dimensional Euclidean Jordan Algebra. + :: - ALGORITHM: + sage: J = OctonionHermitianEJA(2,field=QQ,orthonormalize=False) + sage: I = J.one().to_matrix() + sage: J.trace_inner_product(I, -I) + -2 - For now, we choose a random natural number ``n`` (greater than zero) - and then give you back one of the following: + """ + tr = (X*Y).trace() + if hasattr(tr, 'coefficient'): + # Works for octonions, and has to come first because they + # also have a "real()" method that doesn't return an + # element of the scalar ring. + return tr.coefficient(0) + elif hasattr(tr, 'coefficient_tuple'): + # Works for quaternions. + return tr.coefficient_tuple()[0] - * The cartesian product of the rational numbers ``n`` times; this is - ``QQ^n`` with the Hadamard product. + # Works for real and complex numbers. + return tr.real() - * The Jordan spin algebra on ``QQ^n``. - * The ``n``-by-``n`` rational symmetric matrices with the symmetric - product. - * The ``n``-by-``n`` complex-rational Hermitian matrices embedded - in the space of ``2n``-by-``2n`` real symmetric matrices. +class RealSymmetricEJA(RationalBasisEJA, ConcreteEJA, MatrixEJA): + """ + The rank-n simple EJA consisting of real symmetric n-by-n + matrices, the usual symmetric Jordan product, and the trace inner + product. It has dimension `(n^2 + n)/2` over the reals. - * The ``n``-by-``n`` quaternion-rational Hermitian matrices embedded - in the space of ``4n``-by-``4n`` real symmetric matrices. + SETUP:: - Later this might be extended to return Cartesian products of the - EJAs above. + sage: from mjo.eja.eja_algebra import RealSymmetricEJA - SETUP:: + EXAMPLES:: - sage: from mjo.eja.eja_algebra import random_eja + sage: J = RealSymmetricEJA(2) + sage: b0, b1, b2 = J.gens() + sage: b0*b0 + b0 + sage: b1*b1 + 1/2*b0 + 1/2*b2 + sage: b2*b2 + b2 + + In theory, our "field" can be any subfield of the reals:: + + sage: RealSymmetricEJA(2, field=RDF, check_axioms=True) + Euclidean Jordan algebra of dimension 3 over Real Double Field + sage: RealSymmetricEJA(2, field=RR, check_axioms=True) + Euclidean Jordan algebra of dimension 3 over Real Field with + 53 bits of precision - TESTS:: + TESTS: - sage: random_eja() - Euclidean Jordan algebra of degree... + The dimension of this algebra is `(n^2 + n) / 2`:: - """ + sage: set_random_seed() + sage: n_max = RealSymmetricEJA._max_random_instance_size() + sage: n = ZZ.random_element(1, n_max) + sage: J = RealSymmetricEJA(n) + sage: J.dimension() == (n^2 + n)/2 + True + + The Jordan multiplication is what we think it is:: - # The max_n component lets us choose different upper bounds on the - # value "n" that gets passed to the constructor. This is needed - # because e.g. R^{10} is reasonable to test, while the Hermitian - # 10-by-10 quaternion matrices are not. - (constructor, max_n) = choice([(RealCartesianProductEJA, 6), - (JordanSpinEJA, 6), - (RealSymmetricEJA, 5), - (ComplexHermitianEJA, 4), - (QuaternionHermitianEJA, 3)]) - n = ZZ.random_element(1, max_n) - return constructor(n, field=QQ) + sage: set_random_seed() + sage: J = RealSymmetricEJA.random_instance() + sage: x,y = J.random_elements(2) + sage: actual = (x*y).to_matrix() + sage: X = x.to_matrix() + sage: Y = y.to_matrix() + sage: expected = (X*Y + Y*X)/2 + sage: actual == expected + True + sage: J(expected) == x*y + True + We can change the generator prefix:: + sage: RealSymmetricEJA(3, prefix='q').gens() + (q0, q1, q2, q3, q4, q5) -def _real_symmetric_basis(n, field=QQ): - """ - Return a basis for the space of real symmetric n-by-n matrices. - """ - # The basis of symmetric matrices, as matrices, in their R^(n-by-n) - # coordinates. - S = [] - for i in xrange(n): - for j in xrange(i+1): - Eij = matrix(field, n, lambda k,l: k==i and l==j) - if i == j: - Sij = Eij - else: - # Beware, orthogonal but not normalized! - Sij = Eij + Eij.transpose() - S.append(Sij) - return tuple(S) + We can construct the (trivial) algebra of rank zero:: + sage: RealSymmetricEJA(0) + Euclidean Jordan algebra of dimension 0 over Algebraic Real Field -def _complex_hermitian_basis(n, field=QQ): """ - Returns a basis for the space of complex Hermitian n-by-n matrices. + @staticmethod + def _max_random_instance_size(): + return 4 # Dimension 10 - SETUP:: + @classmethod + def random_instance(cls, **kwargs): + """ + Return a random instance of this type of algebra. + """ + n = ZZ.random_element(cls._max_random_instance_size() + 1) + return cls(n, **kwargs) - sage: from mjo.eja.eja_algebra import _complex_hermitian_basis + def __init__(self, n, field=AA, **kwargs): + # We know this is a valid EJA, but will double-check + # if the user passes check_axioms=True. + if "check_axioms" not in kwargs: kwargs["check_axioms"] = False - TESTS:: + A = MatrixSpace(field, n) + super().__init__(self._denormalized_basis(A), + self.jordan_product, + self.trace_inner_product, + field=field, + **kwargs) - sage: set_random_seed() - sage: n = ZZ.random_element(1,5) - sage: all( M.is_symmetric() for M in _complex_hermitian_basis(n) ) - True + # TODO: this could be factored out somehow, but is left here + # because the MatrixEJA is not presently a subclass of the + # FDEJA class that defines rank() and one(). + self.rank.set_cache(n) + self.one.set_cache(self(A.one())) - """ - F = QuadraticField(-1, 'I') - I = F.gen() - - # This is like the symmetric case, but we need to be careful: - # - # * We want conjugate-symmetry, not just symmetry. - # * The diagonal will (as a result) be real. - # - S = [] - for i in xrange(n): - for j in xrange(i+1): - Eij = matrix(field, n, lambda k,l: k==i and l==j) - if i == j: - Sij = _embed_complex_matrix(Eij) - S.append(Sij) - else: - # Beware, orthogonal but not normalized! The second one - # has a minus because it's conjugated. - Sij_real = _embed_complex_matrix(Eij + Eij.transpose()) - S.append(Sij_real) - Sij_imag = _embed_complex_matrix(I*Eij - I*Eij.transpose()) - S.append(Sij_imag) - return tuple(S) -def _quaternion_hermitian_basis(n, field=QQ): +class ComplexHermitianEJA(RationalBasisEJA, ConcreteEJA, MatrixEJA): """ - Returns a basis for the space of quaternion Hermitian n-by-n matrices. + The rank-n simple EJA consisting of complex Hermitian n-by-n + matrices over the real numbers, the usual symmetric Jordan product, + and the real-part-of-trace inner product. It has dimension `n^2` over + the reals. SETUP:: - sage: from mjo.eja.eja_algebra import _quaternion_hermitian_basis + sage: from mjo.eja.eja_algebra import ComplexHermitianEJA + + EXAMPLES: + + In theory, our "field" can be any subfield of the reals:: - TESTS:: + sage: ComplexHermitianEJA(2, field=RDF, check_axioms=True) + Euclidean Jordan algebra of dimension 4 over Real Double Field + sage: ComplexHermitianEJA(2, field=RR, check_axioms=True) + Euclidean Jordan algebra of dimension 4 over Real Field with + 53 bits of precision + + TESTS: + + The dimension of this algebra is `n^2`:: sage: set_random_seed() - sage: n = ZZ.random_element(1,5) - sage: all( M.is_symmetric() for M in _quaternion_hermitian_basis(n) ) + sage: n_max = ComplexHermitianEJA._max_random_instance_size() + sage: n = ZZ.random_element(1, n_max) + sage: J = ComplexHermitianEJA(n) + sage: J.dimension() == n^2 True + The Jordan multiplication is what we think it is:: + + sage: set_random_seed() + sage: J = ComplexHermitianEJA.random_instance() + sage: x,y = J.random_elements(2) + sage: actual = (x*y).to_matrix() + sage: X = x.to_matrix() + sage: Y = y.to_matrix() + sage: expected = (X*Y + Y*X)/2 + sage: actual == expected + True + sage: J(expected) == x*y + True + + We can change the generator prefix:: + + sage: ComplexHermitianEJA(2, prefix='z').gens() + (z0, z1, z2, z3) + + We can construct the (trivial) algebra of rank zero:: + + sage: ComplexHermitianEJA(0) + Euclidean Jordan algebra of dimension 0 over Algebraic Real Field + """ - Q = QuaternionAlgebra(QQ,-1,-1) - I,J,K = Q.gens() - - # This is like the symmetric case, but we need to be careful: - # - # * We want conjugate-symmetry, not just symmetry. - # * The diagonal will (as a result) be real. - # - S = [] - for i in xrange(n): - for j in xrange(i+1): - Eij = matrix(Q, n, lambda k,l: k==i and l==j) - if i == j: - Sij = _embed_quaternion_matrix(Eij) - S.append(Sij) - else: - # Beware, orthogonal but not normalized! The second, - # third, and fourth ones have a minus because they're - # conjugated. - Sij_real = _embed_quaternion_matrix(Eij + Eij.transpose()) - S.append(Sij_real) - Sij_I = _embed_quaternion_matrix(I*Eij - I*Eij.transpose()) - S.append(Sij_I) - Sij_J = _embed_quaternion_matrix(J*Eij - J*Eij.transpose()) - S.append(Sij_J) - Sij_K = _embed_quaternion_matrix(K*Eij - K*Eij.transpose()) - S.append(Sij_K) - return tuple(S) - - -def _mat2vec(m): - return vector(m.base_ring(), m.list()) - -def _vec2mat(v): - return matrix(v.base_ring(), sqrt(v.degree()), v.list()) - -def _multiplication_table_from_matrix_basis(basis): - """ - At least three of the five simple Euclidean Jordan algebras have the - symmetric multiplication (A,B) |-> (AB + BA)/2, where the - multiplication on the right is matrix multiplication. Given a basis - for the underlying matrix space, this function returns a - multiplication table (obtained by looping through the basis - elements) for an algebra of those matrices. A reordered copy - of the basis is also returned to work around the fact that - the ``span()`` in this function will change the order of the basis - from what we think it is, to... something else. - """ - # In S^2, for example, we nominally have four coordinates even - # though the space is of dimension three only. The vector space V - # is supposed to hold the entire long vector, and the subspace W - # of V will be spanned by the vectors that arise from symmetric - # matrices. Thus for S^2, dim(V) == 4 and dim(W) == 3. - field = basis[0].base_ring() - dimension = basis[0].nrows() - - V = VectorSpace(field, dimension**2) - W = V.span( _mat2vec(s) for s in basis ) - - # Taking the span above reorders our basis (thanks, jerk!) so we - # need to put our "matrix basis" in the same order as the - # (reordered) vector basis. - S = tuple( _vec2mat(b) for b in W.basis() ) - - Qs = [] - for s in S: - # Brute force the multiplication-by-s matrix by looping - # through all elements of the basis and doing the computation - # to find out what the corresponding row should be. BEWARE: - # these multiplication tables won't be symmetric! It therefore - # becomes REALLY IMPORTANT that the underlying algebra - # constructor uses ROW vectors and not COLUMN vectors. That's - # why we're computing rows here and not columns. - Q_rows = [] - for t in S: - this_row = _mat2vec((s*t + t*s)/2) - Q_rows.append(W.coordinates(this_row)) - Q = matrix(field, W.dimension(), Q_rows) - Qs.append(Q) - - return (Qs, S) - - -def _embed_complex_matrix(M): - """ - Embed the n-by-n complex matrix ``M`` into the space of real - matrices of size 2n-by-2n via the map the sends each entry `z = a + - bi` to the block matrix ``[[a,b],[-b,a]]``. + def __init__(self, n, field=AA, **kwargs): + # We know this is a valid EJA, but will double-check + # if the user passes check_axioms=True. + if "check_axioms" not in kwargs: kwargs["check_axioms"] = False + + from mjo.hurwitz import ComplexMatrixAlgebra + A = ComplexMatrixAlgebra(n, scalars=field) + super().__init__(self._denormalized_basis(A), + self.jordan_product, + self.trace_inner_product, + field=field, + **kwargs) + + # TODO: this could be factored out somehow, but is left here + # because the MatrixEJA is not presently a subclass of the + # FDEJA class that defines rank() and one(). + self.rank.set_cache(n) + self.one.set_cache(self(A.one())) + + @staticmethod + def _max_random_instance_size(): + return 3 # Dimension 9 + + @classmethod + def random_instance(cls, **kwargs): + """ + Return a random instance of this type of algebra. + """ + n = ZZ.random_element(cls._max_random_instance_size() + 1) + return cls(n, **kwargs) + + +class QuaternionHermitianEJA(RationalBasisEJA, ConcreteEJA, MatrixEJA): + r""" + The rank-n simple EJA consisting of self-adjoint n-by-n quaternion + matrices, the usual symmetric Jordan product, and the + real-part-of-trace inner product. It has dimension `2n^2 - n` over + the reals. SETUP:: - sage: from mjo.eja.eja_algebra import _embed_complex_matrix + sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA - EXAMPLES:: + EXAMPLES: - sage: F = QuadraticField(-1,'i') - sage: x1 = F(4 - 2*i) - sage: x2 = F(1 + 2*i) - sage: x3 = F(-i) - sage: x4 = F(6) - sage: M = matrix(F,2,[[x1,x2],[x3,x4]]) - sage: _embed_complex_matrix(M) - [ 4 -2| 1 2] - [ 2 4|-2 1] - [-----+-----] - [ 0 -1| 6 0] - [ 1 0| 0 6] + In theory, our "field" can be any subfield of the reals:: + + sage: QuaternionHermitianEJA(2, field=RDF, check_axioms=True) + Euclidean Jordan algebra of dimension 6 over Real Double Field + sage: QuaternionHermitianEJA(2, field=RR, check_axioms=True) + Euclidean Jordan algebra of dimension 6 over Real Field with + 53 bits of precision TESTS: - Embedding is a homomorphism (isomorphism, in fact):: + The dimension of this algebra is `2*n^2 - n`:: sage: set_random_seed() - sage: n = ZZ.random_element(5) - sage: F = QuadraticField(-1, 'i') - sage: X = random_matrix(F, n) - sage: Y = random_matrix(F, n) - sage: actual = _embed_complex_matrix(X) * _embed_complex_matrix(Y) - sage: expected = _embed_complex_matrix(X*Y) + sage: n_max = QuaternionHermitianEJA._max_random_instance_size() + sage: n = ZZ.random_element(1, n_max) + sage: J = QuaternionHermitianEJA(n) + sage: J.dimension() == 2*(n^2) - n + True + + The Jordan multiplication is what we think it is:: + + sage: set_random_seed() + sage: J = QuaternionHermitianEJA.random_instance() + sage: x,y = J.random_elements(2) + sage: actual = (x*y).to_matrix() + sage: X = x.to_matrix() + sage: Y = y.to_matrix() + sage: expected = (X*Y + Y*X)/2 sage: actual == expected True + sage: J(expected) == x*y + True - """ - n = M.nrows() - if M.ncols() != n: - raise ValueError("the matrix 'M' must be square") - field = M.base_ring() - blocks = [] - for z in M.list(): - a = z.real() - b = z.imag() - blocks.append(matrix(field, 2, [[a,b],[-b,a]])) + We can change the generator prefix:: + + sage: QuaternionHermitianEJA(2, prefix='a').gens() + (a0, a1, a2, a3, a4, a5) - # We can drop the imaginaries here. - return matrix.block(field.base_ring(), n, blocks) + We can construct the (trivial) algebra of rank zero:: + sage: QuaternionHermitianEJA(0) + Euclidean Jordan algebra of dimension 0 over Algebraic Real Field -def _unembed_complex_matrix(M): """ - The inverse of _embed_complex_matrix(). + def __init__(self, n, field=AA, **kwargs): + # We know this is a valid EJA, but will double-check + # if the user passes check_axioms=True. + if "check_axioms" not in kwargs: kwargs["check_axioms"] = False + + from mjo.hurwitz import QuaternionMatrixAlgebra + A = QuaternionMatrixAlgebra(n, scalars=field) + super().__init__(self._denormalized_basis(A), + self.jordan_product, + self.trace_inner_product, + field=field, + **kwargs) + + # TODO: this could be factored out somehow, but is left here + # because the MatrixEJA is not presently a subclass of the + # FDEJA class that defines rank() and one(). + self.rank.set_cache(n) + self.one.set_cache(self(A.one())) + + @staticmethod + def _max_random_instance_size(): + r""" + The maximum rank of a random QuaternionHermitianEJA. + """ + return 2 # Dimension 6 + + @classmethod + def random_instance(cls, **kwargs): + """ + Return a random instance of this type of algebra. + """ + n = ZZ.random_element(cls._max_random_instance_size() + 1) + return cls(n, **kwargs) + +class OctonionHermitianEJA(RationalBasisEJA, ConcreteEJA, MatrixEJA): + r""" SETUP:: - sage: from mjo.eja.eja_algebra import (_embed_complex_matrix, - ....: _unembed_complex_matrix) + sage: from mjo.eja.eja_algebra import (FiniteDimensionalEJA, + ....: OctonionHermitianEJA) - EXAMPLES:: + EXAMPLES: - sage: A = matrix(QQ,[ [ 1, 2, 3, 4], - ....: [-2, 1, -4, 3], - ....: [ 9, 10, 11, 12], - ....: [-10, 9, -12, 11] ]) - sage: _unembed_complex_matrix(A) - [ 2*i + 1 4*i + 3] - [ 10*i + 9 12*i + 11] + The 3-by-3 algebra satisfies the axioms of an EJA:: + + sage: OctonionHermitianEJA(3, # long time + ....: field=QQ, # long time + ....: orthonormalize=False, # long time + ....: check_axioms=True) # long time + Euclidean Jordan algebra of dimension 27 over Rational Field + + After a change-of-basis, the 2-by-2 algebra has the same + multiplication table as the ten-dimensional Jordan spin algebra:: + + sage: b = OctonionHermitianEJA._denormalized_basis(2,QQ) + sage: basis = (b[0] + b[9],) + b[1:9] + (b[0] - b[9],) + sage: jp = OctonionHermitianEJA.jordan_product + sage: ip = OctonionHermitianEJA.trace_inner_product + sage: J = FiniteDimensionalEJA(basis, + ....: jp, + ....: ip, + ....: field=QQ, + ....: orthonormalize=False) + sage: J.multiplication_table() + +----++----+----+----+----+----+----+----+----+----+----+ + | * || b0 | b1 | b2 | b3 | b4 | b5 | b6 | b7 | b8 | b9 | + +====++====+====+====+====+====+====+====+====+====+====+ + | b0 || b0 | b1 | b2 | b3 | b4 | b5 | b6 | b7 | b8 | b9 | + +----++----+----+----+----+----+----+----+----+----+----+ + | b1 || b1 | b0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | + +----++----+----+----+----+----+----+----+----+----+----+ + | b2 || b2 | 0 | b0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | + +----++----+----+----+----+----+----+----+----+----+----+ + | b3 || b3 | 0 | 0 | b0 | 0 | 0 | 0 | 0 | 0 | 0 | + +----++----+----+----+----+----+----+----+----+----+----+ + | b4 || b4 | 0 | 0 | 0 | b0 | 0 | 0 | 0 | 0 | 0 | + +----++----+----+----+----+----+----+----+----+----+----+ + | b5 || b5 | 0 | 0 | 0 | 0 | b0 | 0 | 0 | 0 | 0 | + +----++----+----+----+----+----+----+----+----+----+----+ + | b6 || b6 | 0 | 0 | 0 | 0 | 0 | b0 | 0 | 0 | 0 | + +----++----+----+----+----+----+----+----+----+----+----+ + | b7 || b7 | 0 | 0 | 0 | 0 | 0 | 0 | b0 | 0 | 0 | + +----++----+----+----+----+----+----+----+----+----+----+ + | b8 || b8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | b0 | 0 | + +----++----+----+----+----+----+----+----+----+----+----+ + | b9 || b9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | b0 | + +----++----+----+----+----+----+----+----+----+----+----+ TESTS: - Unembedding is the inverse of embedding:: - - sage: set_random_seed() - sage: F = QuadraticField(-1, 'i') - sage: M = random_matrix(F, 3) - sage: _unembed_complex_matrix(_embed_complex_matrix(M)) == M - True + We can actually construct the 27-dimensional Albert algebra, + and we get the right unit element if we recompute it:: + + sage: J = OctonionHermitianEJA(3, # long time + ....: field=QQ, # long time + ....: orthonormalize=False) # long time + sage: J.one.clear_cache() # long time + sage: J.one() # long time + b0 + b9 + b26 + sage: J.one().to_matrix() # long time + +----+----+----+ + | e0 | 0 | 0 | + +----+----+----+ + | 0 | e0 | 0 | + +----+----+----+ + | 0 | 0 | e0 | + +----+----+----+ + + The 2-by-2 algebra is isomorphic to the ten-dimensional Jordan + spin algebra, but just to be sure, we recompute its rank:: + + sage: J = OctonionHermitianEJA(2, # long time + ....: field=QQ, # long time + ....: orthonormalize=False) # long time + sage: J.rank.clear_cache() # long time + sage: J.rank() # long time + 2 """ - n = ZZ(M.nrows()) - if M.ncols() != n: - raise ValueError("the matrix 'M' must be square") - if not n.mod(2).is_zero(): - raise ValueError("the matrix 'M' must be a complex embedding") - - F = QuadraticField(-1, 'i') - i = F.gen() - - # Go top-left to bottom-right (reading order), converting every - # 2-by-2 block we see to a single complex element. - elements = [] - for k in xrange(n/2): - for j in xrange(n/2): - submat = M[2*k:2*k+2,2*j:2*j+2] - if submat[0,0] != submat[1,1]: - raise ValueError('bad on-diagonal submatrix') - if submat[0,1] != -submat[1,0]: - raise ValueError('bad off-diagonal submatrix') - z = submat[0,0] + submat[0,1]*i - elements.append(z) - - return matrix(F, n/2, elements) - - -def _embed_quaternion_matrix(M): - """ - Embed the n-by-n quaternion matrix ``M`` into the space of real - matrices of size 4n-by-4n by first sending each quaternion entry - `z = a + bi + cj + dk` to the block-complex matrix - ``[[a + bi, c+di],[-c + di, a-bi]]`, and then embedding those into - a real matrix. + @staticmethod + def _max_random_instance_size(): + r""" + The maximum rank of a random QuaternionHermitianEJA. + """ + return 1 # Dimension 1 + + @classmethod + def random_instance(cls, **kwargs): + """ + Return a random instance of this type of algebra. + """ + n = ZZ.random_element(cls._max_random_instance_size() + 1) + return cls(n, **kwargs) + + def __init__(self, n, field=AA, **kwargs): + if n > 3: + # Otherwise we don't get an EJA. + raise ValueError("n cannot exceed 3") + + # We know this is a valid EJA, but will double-check + # if the user passes check_axioms=True. + if "check_axioms" not in kwargs: kwargs["check_axioms"] = False + + from mjo.hurwitz import OctonionMatrixAlgebra + A = OctonionMatrixAlgebra(n, scalars=field) + super().__init__(self._denormalized_basis(A), + self.jordan_product, + self.trace_inner_product, + field=field, + **kwargs) + + # TODO: this could be factored out somehow, but is left here + # because the MatrixEJA is not presently a subclass of the + # FDEJA class that defines rank() and one(). + self.rank.set_cache(n) + self.one.set_cache(self(A.one())) + + +class AlbertEJA(OctonionHermitianEJA): + r""" + The Albert algebra is the algebra of three-by-three Hermitian + matrices whose entries are octonions. SETUP:: - sage: from mjo.eja.eja_algebra import _embed_quaternion_matrix + sage: from mjo.eja.eja_algebra import AlbertEJA EXAMPLES:: - sage: Q = QuaternionAlgebra(QQ,-1,-1) - sage: i,j,k = Q.gens() - sage: x = 1 + 2*i + 3*j + 4*k - sage: M = matrix(Q, 1, [[x]]) - sage: _embed_quaternion_matrix(M) - [ 1 2 3 4] - [-2 1 -4 3] - [-3 4 1 -2] - [-4 -3 2 1] + sage: AlbertEJA(field=QQ, orthonormalize=False) + Euclidean Jordan algebra of dimension 27 over Rational Field + sage: AlbertEJA() # long time + Euclidean Jordan algebra of dimension 27 over Algebraic Real Field - Embedding is a homomorphism (isomorphism, in fact):: + """ + def __init__(self, *args, **kwargs): + super().__init__(3, *args, **kwargs) - sage: set_random_seed() - sage: n = ZZ.random_element(5) - sage: Q = QuaternionAlgebra(QQ,-1,-1) - sage: X = random_matrix(Q, n) - sage: Y = random_matrix(Q, n) - sage: actual = _embed_quaternion_matrix(X)*_embed_quaternion_matrix(Y) - sage: expected = _embed_quaternion_matrix(X*Y) - sage: actual == expected - True +class HadamardEJA(RationalBasisEJA, ConcreteEJA): """ - quaternions = M.base_ring() - n = M.nrows() - if M.ncols() != n: - raise ValueError("the matrix 'M' must be square") - - F = QuadraticField(-1, 'i') - i = F.gen() - - blocks = [] - for z in M.list(): - t = z.coefficient_tuple() - a = t[0] - b = t[1] - c = t[2] - d = t[3] - cplx_matrix = matrix(F, 2, [[ a + b*i, c + d*i], - [-c + d*i, a - b*i]]) - blocks.append(_embed_complex_matrix(cplx_matrix)) - - # We should have real entries by now, so use the realest field - # we've got for the return value. - return matrix.block(quaternions.base_ring(), n, blocks) - - -def _unembed_quaternion_matrix(M): - """ - The inverse of _embed_quaternion_matrix(). + Return the Euclidean Jordan algebra on `R^n` with the Hadamard + (pointwise real-number multiplication) Jordan product and the + usual inner-product. + + This is nothing more than the Cartesian product of ``n`` copies of + the one-dimensional Jordan spin algebra, and is the most common + example of a non-simple Euclidean Jordan algebra. SETUP:: - sage: from mjo.eja.eja_algebra import (_embed_quaternion_matrix, - ....: _unembed_quaternion_matrix) + sage: from mjo.eja.eja_algebra import HadamardEJA - EXAMPLES:: + EXAMPLES: - sage: M = matrix(QQ, [[ 1, 2, 3, 4], - ....: [-2, 1, -4, 3], - ....: [-3, 4, 1, -2], - ....: [-4, -3, 2, 1]]) - sage: _unembed_quaternion_matrix(M) - [1 + 2*i + 3*j + 4*k] + This multiplication table can be verified by hand:: - TESTS: + sage: J = HadamardEJA(3) + sage: b0,b1,b2 = J.gens() + sage: b0*b0 + b0 + sage: b0*b1 + 0 + sage: b0*b2 + 0 + sage: b1*b1 + b1 + sage: b1*b2 + 0 + sage: b2*b2 + b2 - Unembedding is the inverse of embedding:: + TESTS: - sage: set_random_seed() - sage: Q = QuaternionAlgebra(QQ, -1, -1) - sage: M = random_matrix(Q, 3) - sage: _unembed_quaternion_matrix(_embed_quaternion_matrix(M)) == M - True + We can change the generator prefix:: + sage: HadamardEJA(3, prefix='r').gens() + (r0, r1, r2) """ - n = ZZ(M.nrows()) - if M.ncols() != n: - raise ValueError("the matrix 'M' must be square") - if not n.mod(4).is_zero(): - raise ValueError("the matrix 'M' must be a complex embedding") - - Q = QuaternionAlgebra(QQ,-1,-1) - i,j,k = Q.gens() - - # Go top-left to bottom-right (reading order), converting every - # 4-by-4 block we see to a 2-by-2 complex block, to a 1-by-1 - # quaternion block. - elements = [] - for l in xrange(n/4): - for m in xrange(n/4): - submat = _unembed_complex_matrix(M[4*l:4*l+4,4*m:4*m+4]) - if submat[0,0] != submat[1,1].conjugate(): - raise ValueError('bad on-diagonal submatrix') - if submat[0,1] != -submat[1,0].conjugate(): - raise ValueError('bad off-diagonal submatrix') - z = submat[0,0].real() + submat[0,0].imag()*i - z += submat[0,1].real()*j + submat[0,1].imag()*k - elements.append(z) - - return matrix(Q, n/4, elements) - - -# The usual inner product on R^n. -def _usual_ip(x,y): - return x.vector().inner_product(y.vector()) - -# The inner product used for the real symmetric simple EJA. -# We keep it as a separate function because e.g. the complex -# algebra uses the same inner product, except divided by 2. -def _matrix_ip(X,Y): - X_mat = X.natural_representation() - Y_mat = Y.natural_representation() - return (X_mat*Y_mat).trace() - - -class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra): - """ - The rank-n simple EJA consisting of real symmetric n-by-n - matrices, the usual symmetric Jordan product, and the trace inner - product. It has dimension `(n^2 + n)/2` over the reals. + def __init__(self, n, field=AA, **kwargs): + if n == 0: + jordan_product = lambda x,y: x + inner_product = lambda x,y: x + else: + def jordan_product(x,y): + P = x.parent() + return P( xi*yi for (xi,yi) in zip(x,y) ) + + def inner_product(x,y): + return (x.T*y)[0,0] + + # New defaults for keyword arguments. Don't orthonormalize + # because our basis is already orthonormal with respect to our + # inner-product. Don't check the axioms, because we know this + # is a valid EJA... but do double-check if the user passes + # check_axioms=True. Note: we DON'T override the "check_field" + # default here, because the user can pass in a field! + if "orthonormalize" not in kwargs: kwargs["orthonormalize"] = False + if "check_axioms" not in kwargs: kwargs["check_axioms"] = False + + column_basis = tuple( b.column() + for b in FreeModule(field, n).basis() ) + super().__init__(column_basis, + jordan_product, + inner_product, + field=field, + associative=True, + **kwargs) + self.rank.set_cache(n) + + if n == 0: + self.one.set_cache( self.zero() ) + else: + self.one.set_cache( sum(self.gens()) ) + + @staticmethod + def _max_random_instance_size(): + r""" + The maximum dimension of a random HadamardEJA. + """ + return 5 + + @classmethod + def random_instance(cls, **kwargs): + """ + Return a random instance of this type of algebra. + """ + n = ZZ.random_element(cls._max_random_instance_size() + 1) + return cls(n, **kwargs) + + +class BilinearFormEJA(RationalBasisEJA, ConcreteEJA): + r""" + The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)`` + with the half-trace inner product and jordan product ``x*y = + (,y_bar>, x0*y_bar + y0*x_bar)`` where `B = 1 \times B22` is + a symmetric positive-definite "bilinear form" matrix. Its + dimension is the size of `B`, and it has rank two in dimensions + larger than two. It reduces to the ``JordanSpinEJA`` when `B` is + the identity matrix of order ``n``. + + We insist that the one-by-one upper-left identity block of `B` be + passed in as well so that we can be passed a matrix of size zero + to construct a trivial algebra. SETUP:: - sage: from mjo.eja.eja_algebra import RealSymmetricEJA + sage: from mjo.eja.eja_algebra import (BilinearFormEJA, + ....: JordanSpinEJA) - EXAMPLES:: + EXAMPLES: - sage: J = RealSymmetricEJA(2) - sage: e0, e1, e2 = J.gens() - sage: e0*e0 - e0 - sage: e1*e1 - e0 + e2 - sage: e2*e2 - e2 + When no bilinear form is specified, the identity matrix is used, + and the resulting algebra is the Jordan spin algebra:: + + sage: B = matrix.identity(AA,3) + sage: J0 = BilinearFormEJA(B) + sage: J1 = JordanSpinEJA(3) + sage: J0.multiplication_table() == J0.multiplication_table() + True + + An error is raised if the matrix `B` does not correspond to a + positive-definite bilinear form:: + + sage: B = matrix.random(QQ,2,3) + sage: J = BilinearFormEJA(B) + Traceback (most recent call last): + ... + ValueError: bilinear form is not positive-definite + sage: B = matrix.zero(QQ,3) + sage: J = BilinearFormEJA(B) + Traceback (most recent call last): + ... + ValueError: bilinear form is not positive-definite TESTS: - The degree of this algebra is `(n^2 + n) / 2`:: + We can create a zero-dimensional algebra:: - sage: set_random_seed() - sage: n = ZZ.random_element(1,5) - sage: J = RealSymmetricEJA(n) - sage: J.degree() == (n^2 + n)/2 - True + sage: B = matrix.identity(AA,0) + sage: J = BilinearFormEJA(B) + sage: J.basis() + Finite family {} - The Jordan multiplication is what we think it is:: + We can check the multiplication condition given in the Jordan, von + Neumann, and Wigner paper (and also discussed on my "On the + symmetry..." paper). Note that this relies heavily on the standard + choice of basis, as does anything utilizing the bilinear form + matrix. We opt not to orthonormalize the basis, because if we + did, we would have to normalize the `s_{i}` in a similar manner:: sage: set_random_seed() - sage: n = ZZ.random_element(1,5) - sage: J = RealSymmetricEJA(n) - sage: x = J.random_element() - sage: y = J.random_element() - sage: actual = (x*y).natural_representation() - sage: X = x.natural_representation() - sage: Y = y.natural_representation() - sage: expected = (X*Y + Y*X)/2 + sage: n = ZZ.random_element(5) + sage: M = matrix.random(QQ, max(0,n-1), algorithm='unimodular') + sage: B11 = matrix.identity(QQ,1) + sage: B22 = M.transpose()*M + sage: B = block_matrix(2,2,[ [B11,0 ], + ....: [0, B22 ] ]) + sage: J = BilinearFormEJA(B, orthonormalize=False) + sage: eis = VectorSpace(M.base_ring(), M.ncols()).basis() + sage: V = J.vector_space() + sage: sis = [ J( V([0] + (M.inverse()*ei).list()).column() ) + ....: for ei in eis ] + sage: actual = [ sis[i]*sis[j] + ....: for i in range(n-1) + ....: for j in range(n-1) ] + sage: expected = [ J.one() if i == j else J.zero() + ....: for i in range(n-1) + ....: for j in range(n-1) ] sage: actual == expected True - sage: J(expected) == x*y - True """ + def __init__(self, B, field=AA, **kwargs): + # The matrix "B" is supplied by the user in most cases, + # so it makes sense to check whether or not its positive- + # definite unless we are specifically asked not to... + if ("check_axioms" not in kwargs) or kwargs["check_axioms"]: + if not B.is_positive_definite(): + raise ValueError("bilinear form is not positive-definite") + + # However, all of the other data for this EJA is computed + # by us in manner that guarantees the axioms are + # satisfied. So, again, unless we are specifically asked to + # verify things, we'll skip the rest of the checks. + if "check_axioms" not in kwargs: kwargs["check_axioms"] = False + + def inner_product(x,y): + return (y.T*B*x)[0,0] + + def jordan_product(x,y): + P = x.parent() + x0 = x[0,0] + xbar = x[1:,0] + y0 = y[0,0] + ybar = y[1:,0] + z0 = inner_product(y,x) + zbar = y0*xbar + x0*ybar + return P([z0] + zbar.list()) + + n = B.nrows() + column_basis = tuple( b.column() + for b in FreeModule(field, n).basis() ) + + # TODO: I haven't actually checked this, but it seems legit. + associative = False + if n <= 2: + associative = True + + super().__init__(column_basis, + jordan_product, + inner_product, + field=field, + associative=associative, + **kwargs) + + # The rank of this algebra is two, unless we're in a + # one-dimensional ambient space (because the rank is bounded + # by the ambient dimension). + self.rank.set_cache(min(n,2)) + + if n == 0: + self.one.set_cache( self.zero() ) + else: + self.one.set_cache( self.monomial(0) ) + @staticmethod - def __classcall_private__(cls, n, field=QQ): - S = _real_symmetric_basis(n, field=field) - (Qs, T) = _multiplication_table_from_matrix_basis(S) + def _max_random_instance_size(): + r""" + The maximum dimension of a random BilinearFormEJA. + """ + return 5 - fdeja = super(RealSymmetricEJA, cls) - return fdeja.__classcall_private__(cls, - field, - Qs, - rank=n, - natural_basis=T) + @classmethod + def random_instance(cls, **kwargs): + """ + Return a random instance of this algebra. + """ + n = ZZ.random_element(cls._max_random_instance_size() + 1) + if n.is_zero(): + B = matrix.identity(ZZ, n) + return cls(B, **kwargs) - def inner_product(self, x, y): - return _matrix_ip(x,y) + B11 = matrix.identity(ZZ, 1) + M = matrix.random(ZZ, n-1) + I = matrix.identity(ZZ, n-1) + alpha = ZZ.zero() + while alpha.is_zero(): + alpha = ZZ.random_element().abs() + B22 = M.transpose()*M + alpha*I + from sage.matrix.special import block_matrix + B = block_matrix(2,2, [ [B11, ZZ(0) ], + [ZZ(0), B22 ] ]) -class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra): + return cls(B, **kwargs) + + +class JordanSpinEJA(BilinearFormEJA): """ - The rank-n simple EJA consisting of complex Hermitian n-by-n - matrices over the real numbers, the usual symmetric Jordan product, - and the real-part-of-trace inner product. It has dimension `n^2` over + The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)`` + with the usual inner product and jordan product ``x*y = + (, x0*y_bar + y0*x_bar)``. It has dimension `n` over the reals. SETUP:: - sage: from mjo.eja.eja_algebra import ComplexHermitianEJA + sage: from mjo.eja.eja_algebra import JordanSpinEJA - TESTS: + EXAMPLES: - The degree of this algebra is `n^2`:: + This multiplication table can be verified by hand:: - sage: set_random_seed() - sage: n = ZZ.random_element(1,5) - sage: J = ComplexHermitianEJA(n) - sage: J.degree() == n^2 - True + sage: J = JordanSpinEJA(4) + sage: b0,b1,b2,b3 = J.gens() + sage: b0*b0 + b0 + sage: b0*b1 + b1 + sage: b0*b2 + b2 + sage: b0*b3 + b3 + sage: b1*b2 + 0 + sage: b1*b3 + 0 + sage: b2*b3 + 0 - The Jordan multiplication is what we think it is:: + We can change the generator prefix:: - sage: set_random_seed() - sage: n = ZZ.random_element(1,5) - sage: J = ComplexHermitianEJA(n) - sage: x = J.random_element() - sage: y = J.random_element() - sage: actual = (x*y).natural_representation() - sage: X = x.natural_representation() - sage: Y = y.natural_representation() - sage: expected = (X*Y + Y*X)/2 - sage: actual == expected - True - sage: J(expected) == x*y - True + sage: JordanSpinEJA(2, prefix='B').gens() + (B0, B1) + + TESTS: + + Ensure that we have the usual inner product on `R^n`:: + + sage: set_random_seed() + sage: J = JordanSpinEJA.random_instance() + sage: x,y = J.random_elements(2) + sage: actual = x.inner_product(y) + sage: expected = x.to_vector().inner_product(y.to_vector()) + sage: actual == expected + True """ + def __init__(self, n, *args, **kwargs): + # This is a special case of the BilinearFormEJA with the + # identity matrix as its bilinear form. + B = matrix.identity(ZZ, n) + + # Don't orthonormalize because our basis is already + # orthonormal with respect to our inner-product. + if "orthonormalize" not in kwargs: kwargs["orthonormalize"] = False + + # But also don't pass check_field=False here, because the user + # can pass in a field! + super().__init__(B, *args, **kwargs) + @staticmethod - def __classcall_private__(cls, n, field=QQ): - S = _complex_hermitian_basis(n) - (Qs, T) = _multiplication_table_from_matrix_basis(S) + def _max_random_instance_size(): + r""" + The maximum dimension of a random JordanSpinEJA. + """ + return 5 - fdeja = super(ComplexHermitianEJA, cls) - return fdeja.__classcall_private__(cls, - field, - Qs, - rank=n, - natural_basis=T) + @classmethod + def random_instance(cls, **kwargs): + """ + Return a random instance of this type of algebra. - def inner_product(self, x, y): - # Since a+bi on the diagonal is represented as - # - # a + bi = [ a b ] - # [ -b a ], - # - # we'll double-count the "a" entries if we take the trace of - # the embedding. - return _matrix_ip(x,y)/2 + Needed here to override the implementation for ``BilinearFormEJA``. + """ + n = ZZ.random_element(cls._max_random_instance_size() + 1) + return cls(n, **kwargs) -class QuaternionHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra): +class TrivialEJA(RationalBasisEJA, ConcreteEJA): """ - The rank-n simple EJA consisting of self-adjoint n-by-n quaternion - matrices, the usual symmetric Jordan product, and the - real-part-of-trace inner product. It has dimension `2n^2 - n` over - the reals. + The trivial Euclidean Jordan algebra consisting of only a zero element. SETUP:: - sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA + sage: from mjo.eja.eja_algebra import TrivialEJA - TESTS: + EXAMPLES:: + + sage: J = TrivialEJA() + sage: J.dimension() + 0 + sage: J.zero() + 0 + sage: J.one() + 0 + sage: 7*J.one()*12*J.one() + 0 + sage: J.one().inner_product(J.one()) + 0 + sage: J.one().norm() + 0 + sage: J.one().subalgebra_generated_by() + Euclidean Jordan algebra of dimension 0 over Algebraic Real Field + sage: J.rank() + 0 + + """ + def __init__(self, **kwargs): + jordan_product = lambda x,y: x + inner_product = lambda x,y: 0 + basis = () + + # New defaults for keyword arguments + if "orthonormalize" not in kwargs: kwargs["orthonormalize"] = False + if "check_axioms" not in kwargs: kwargs["check_axioms"] = False + + super().__init__(basis, + jordan_product, + inner_product, + associative=True, + **kwargs) + + # The rank is zero using my definition, namely the dimension of the + # largest subalgebra generated by any element. + self.rank.set_cache(0) + self.one.set_cache( self.zero() ) + + @classmethod + def random_instance(cls, **kwargs): + # We don't take a "size" argument so the superclass method is + # inappropriate for us. + return cls(**kwargs) + + +class CartesianProductEJA(FiniteDimensionalEJA): + r""" + The external (orthogonal) direct sum of two or more Euclidean + Jordan algebras. Every Euclidean Jordan algebra decomposes into an + orthogonal direct sum of simple Euclidean Jordan algebras which is + then isometric to a Cartesian product, so no generality is lost by + providing only this construction. + + SETUP:: - The degree of this algebra is `n^2`:: + sage: from mjo.eja.eja_algebra import (random_eja, + ....: CartesianProductEJA, + ....: HadamardEJA, + ....: JordanSpinEJA, + ....: RealSymmetricEJA) + + EXAMPLES: + + The Jordan product is inherited from our factors and implemented by + our CombinatorialFreeModule Cartesian product superclass:: sage: set_random_seed() - sage: n = ZZ.random_element(1,5) - sage: J = QuaternionHermitianEJA(n) - sage: J.degree() == 2*(n^2) - n + sage: J1 = HadamardEJA(2) + sage: J2 = RealSymmetricEJA(2) + sage: J = cartesian_product([J1,J2]) + sage: x,y = J.random_elements(2) + sage: x*y in J True - The Jordan multiplication is what we think it is:: + The ability to retrieve the original factors is implemented by our + CombinatorialFreeModule Cartesian product superclass:: + + sage: J1 = HadamardEJA(2, field=QQ) + sage: J2 = JordanSpinEJA(3, field=QQ) + sage: J = cartesian_product([J1,J2]) + sage: J.cartesian_factors() + (Euclidean Jordan algebra of dimension 2 over Rational Field, + Euclidean Jordan algebra of dimension 3 over Rational Field) + + You can provide more than two factors:: + + sage: J1 = HadamardEJA(2) + sage: J2 = JordanSpinEJA(3) + sage: J3 = RealSymmetricEJA(3) + sage: cartesian_product([J1,J2,J3]) + Euclidean Jordan algebra of dimension 2 over Algebraic Real + Field (+) Euclidean Jordan algebra of dimension 3 over Algebraic + Real Field (+) Euclidean Jordan algebra of dimension 6 over + Algebraic Real Field + + Rank is additive on a Cartesian product:: + + sage: J1 = HadamardEJA(1) + sage: J2 = RealSymmetricEJA(2) + sage: J = cartesian_product([J1,J2]) + sage: J1.rank.clear_cache() + sage: J2.rank.clear_cache() + sage: J.rank.clear_cache() + sage: J.rank() + 3 + sage: J.rank() == J1.rank() + J2.rank() + True - sage: set_random_seed() - sage: n = ZZ.random_element(1,5) - sage: J = QuaternionHermitianEJA(n) - sage: x = J.random_element() - sage: y = J.random_element() - sage: actual = (x*y).natural_representation() - sage: X = x.natural_representation() - sage: Y = y.natural_representation() - sage: expected = (X*Y + Y*X)/2 - sage: actual == expected + The same rank computation works over the rationals, with whatever + basis you like:: + + sage: J1 = HadamardEJA(1, field=QQ, orthonormalize=False) + sage: J2 = RealSymmetricEJA(2, field=QQ, orthonormalize=False) + sage: J = cartesian_product([J1,J2]) + sage: J1.rank.clear_cache() + sage: J2.rank.clear_cache() + sage: J.rank.clear_cache() + sage: J.rank() + 3 + sage: J.rank() == J1.rank() + J2.rank() True - sage: J(expected) == x*y + + The product algebra will be associative if and only if all of its + components are associative:: + + sage: J1 = HadamardEJA(2) + sage: J1.is_associative() + True + sage: J2 = HadamardEJA(3) + sage: J2.is_associative() + True + sage: J3 = RealSymmetricEJA(3) + sage: J3.is_associative() + False + sage: CP1 = cartesian_product([J1,J2]) + sage: CP1.is_associative() + True + sage: CP2 = cartesian_product([J1,J3]) + sage: CP2.is_associative() + False + + Cartesian products of Cartesian products work:: + + sage: J1 = JordanSpinEJA(1) + sage: J2 = JordanSpinEJA(1) + sage: J3 = JordanSpinEJA(1) + sage: J = cartesian_product([J1,cartesian_product([J2,J3])]) + sage: J.multiplication_table() + +----++----+----+----+ + | * || b0 | b1 | b2 | + +====++====+====+====+ + | b0 || b0 | 0 | 0 | + +----++----+----+----+ + | b1 || 0 | b1 | 0 | + +----++----+----+----+ + | b2 || 0 | 0 | b2 | + +----++----+----+----+ + sage: HadamardEJA(3).multiplication_table() + +----++----+----+----+ + | * || b0 | b1 | b2 | + +====++====+====+====+ + | b0 || b0 | 0 | 0 | + +----++----+----+----+ + | b1 || 0 | b1 | 0 | + +----++----+----+----+ + | b2 || 0 | 0 | b2 | + +----++----+----+----+ + + TESTS: + + All factors must share the same base field:: + + sage: J1 = HadamardEJA(2, field=QQ) + sage: J2 = RealSymmetricEJA(2) + sage: CartesianProductEJA((J1,J2)) + Traceback (most recent call last): + ... + ValueError: all factors must share the same base field + + The cached unit element is the same one that would be computed:: + + sage: set_random_seed() # long time + sage: J1 = random_eja() # long time + sage: J2 = random_eja() # long time + sage: J = cartesian_product([J1,J2]) # long time + sage: actual = J.one() # long time + sage: J.one.clear_cache() # long time + sage: expected = J.one() # long time + sage: actual == expected # long time True """ - @staticmethod - def __classcall_private__(cls, n, field=QQ): - S = _quaternion_hermitian_basis(n) - (Qs, T) = _multiplication_table_from_matrix_basis(S) + Element = FiniteDimensionalEJAElement - fdeja = super(QuaternionHermitianEJA, cls) - return fdeja.__classcall_private__(cls, - field, - Qs, - rank=n, - natural_basis=T) - def inner_product(self, x, y): - # Since a+bi+cj+dk on the diagonal is represented as - # - # a + bi +cj + dk = [ a b c d] - # [ -b a -d c] - # [ -c d a -b] - # [ -d -c b a], + def __init__(self, factors, **kwargs): + m = len(factors) + if m == 0: + return TrivialEJA() + + self._sets = factors + + field = factors[0].base_ring() + if not all( J.base_ring() == field for J in factors ): + raise ValueError("all factors must share the same base field") + + associative = all( f.is_associative() for f in factors ) + + MS = self.matrix_space() + basis = [] + zero = MS.zero() + for i in range(m): + for b in factors[i].matrix_basis(): + z = list(zero) + z[i] = b + basis.append(z) + + basis = tuple( MS(b) for b in basis ) + + # Define jordan/inner products that operate on that matrix_basis. + def jordan_product(x,y): + return MS(tuple( + (factors[i](x[i])*factors[i](y[i])).to_matrix() + for i in range(m) + )) + + def inner_product(x, y): + return sum( + factors[i](x[i]).inner_product(factors[i](y[i])) + for i in range(m) + ) + + # There's no need to check the field since it already came + # from an EJA. Likewise the axioms are guaranteed to be + # satisfied, unless the guy writing this class sucks. # - # we'll quadruple-count the "a" entries if we take the trace of - # the embedding. - return _matrix_ip(x,y)/4 + # If you want the basis to be orthonormalized, orthonormalize + # the factors. + FiniteDimensionalEJA.__init__(self, + basis, + jordan_product, + inner_product, + field=field, + orthonormalize=False, + associative=associative, + cartesian_product=True, + check_field=False, + check_axioms=False) + + ones = tuple(J.one().to_matrix() for J in factors) + self.one.set_cache(self(ones)) + self.rank.set_cache(sum(J.rank() for J in factors)) + + def cartesian_factors(self): + # Copy/pasted from CombinatorialFreeModule_CartesianProduct. + return self._sets + + def cartesian_factor(self, i): + r""" + Return the ``i``th factor of this algebra. + """ + return self._sets[i] + def _repr_(self): + # Copy/pasted from CombinatorialFreeModule_CartesianProduct. + from sage.categories.cartesian_product import cartesian_product + return cartesian_product.symbol.join("%s" % factor + for factor in self._sets) + + def matrix_space(self): + r""" + Return the space that our matrix basis lives in as a Cartesian + product. -class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra): - """ - The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)`` - with the usual inner product and jordan product ``x*y = - (, x0*y_bar + y0*x_bar)``. It has dimension `n` over - the reals. + We don't simply use the ``cartesian_product()`` functor here + because it acts differently on SageMath MatrixSpaces and our + custom MatrixAlgebras, which are CombinatorialFreeModules. We + always want the result to be represented (and indexed) as + an ordered tuple. + + SETUP:: + + sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA, + ....: HadamardEJA, + ....: OctonionHermitianEJA, + ....: RealSymmetricEJA) + + EXAMPLES:: + + sage: J1 = HadamardEJA(1) + sage: J2 = RealSymmetricEJA(2) + sage: J = cartesian_product([J1,J2]) + sage: J.matrix_space() + The Cartesian product of (Full MatrixSpace of 1 by 1 dense + matrices over Algebraic Real Field, Full MatrixSpace of 2 + by 2 dense matrices over Algebraic Real Field) + + :: + + sage: J1 = ComplexHermitianEJA(1) + sage: J2 = ComplexHermitianEJA(1) + sage: J = cartesian_product([J1,J2]) + sage: J.one().to_matrix()[0] + [1 0] + [0 1] + sage: J.one().to_matrix()[1] + [1 0] + [0 1] + + :: + + sage: J1 = OctonionHermitianEJA(1) + sage: J2 = OctonionHermitianEJA(1) + sage: J = cartesian_product([J1,J2]) + sage: J.one().to_matrix()[0] + +----+ + | e0 | + +----+ + sage: J.one().to_matrix()[1] + +----+ + | e0 | + +----+ + + """ + scalars = self.cartesian_factor(0).base_ring() + + # This category isn't perfect, but is good enough for what we + # need to do. + cat = MagmaticAlgebras(scalars).FiniteDimensional().WithBasis() + cat = cat.Unital().CartesianProducts() + factors = tuple( J.matrix_space() for J in self.cartesian_factors() ) + + from sage.sets.cartesian_product import CartesianProduct + return CartesianProduct(factors, cat) + + + @cached_method + def cartesian_projection(self, i): + r""" + SETUP:: + + sage: from mjo.eja.eja_algebra import (random_eja, + ....: JordanSpinEJA, + ....: HadamardEJA, + ....: RealSymmetricEJA, + ....: ComplexHermitianEJA) + + EXAMPLES: + + The projection morphisms are Euclidean Jordan algebra + operators:: + + sage: J1 = HadamardEJA(2) + sage: J2 = RealSymmetricEJA(2) + sage: J = cartesian_product([J1,J2]) + sage: J.cartesian_projection(0) + Linear operator between finite-dimensional Euclidean Jordan + algebras represented by the matrix: + [1 0 0 0 0] + [0 1 0 0 0] + Domain: Euclidean Jordan algebra of dimension 2 over Algebraic + Real Field (+) Euclidean Jordan algebra of dimension 3 over + Algebraic Real Field + Codomain: Euclidean Jordan algebra of dimension 2 over Algebraic + Real Field + sage: J.cartesian_projection(1) + Linear operator between finite-dimensional Euclidean Jordan + algebras represented by the matrix: + [0 0 1 0 0] + [0 0 0 1 0] + [0 0 0 0 1] + Domain: Euclidean Jordan algebra of dimension 2 over Algebraic + Real Field (+) Euclidean Jordan algebra of dimension 3 over + Algebraic Real Field + Codomain: Euclidean Jordan algebra of dimension 3 over Algebraic + Real Field + + The projections work the way you'd expect on the vector + representation of an element:: + + sage: J1 = JordanSpinEJA(2) + sage: J2 = ComplexHermitianEJA(2) + sage: J = cartesian_product([J1,J2]) + sage: pi_left = J.cartesian_projection(0) + sage: pi_right = J.cartesian_projection(1) + sage: pi_left(J.one()).to_vector() + (1, 0) + sage: pi_right(J.one()).to_vector() + (1, 0, 0, 1) + sage: J.one().to_vector() + (1, 0, 1, 0, 0, 1) + + TESTS: + + The answer never changes:: + + sage: set_random_seed() + sage: J1 = random_eja() + sage: J2 = random_eja() + sage: J = cartesian_product([J1,J2]) + sage: P0 = J.cartesian_projection(0) + sage: P1 = J.cartesian_projection(0) + sage: P0 == P1 + True + + """ + offset = sum( self.cartesian_factor(k).dimension() + for k in range(i) ) + Ji = self.cartesian_factor(i) + Pi = self._module_morphism(lambda j: Ji.monomial(j - offset), + codomain=Ji) + + return FiniteDimensionalEJAOperator(self,Ji,Pi.matrix()) + + @cached_method + def cartesian_embedding(self, i): + r""" + SETUP:: + + sage: from mjo.eja.eja_algebra import (random_eja, + ....: JordanSpinEJA, + ....: HadamardEJA, + ....: RealSymmetricEJA) + + EXAMPLES: + + The embedding morphisms are Euclidean Jordan algebra + operators:: + + sage: J1 = HadamardEJA(2) + sage: J2 = RealSymmetricEJA(2) + sage: J = cartesian_product([J1,J2]) + sage: J.cartesian_embedding(0) + Linear operator between finite-dimensional Euclidean Jordan + algebras represented by the matrix: + [1 0] + [0 1] + [0 0] + [0 0] + [0 0] + Domain: Euclidean Jordan algebra of dimension 2 over + Algebraic Real Field + Codomain: Euclidean Jordan algebra of dimension 2 over + Algebraic Real Field (+) Euclidean Jordan algebra of + dimension 3 over Algebraic Real Field + sage: J.cartesian_embedding(1) + Linear operator between finite-dimensional Euclidean Jordan + algebras represented by the matrix: + [0 0 0] + [0 0 0] + [1 0 0] + [0 1 0] + [0 0 1] + Domain: Euclidean Jordan algebra of dimension 3 over + Algebraic Real Field + Codomain: Euclidean Jordan algebra of dimension 2 over + Algebraic Real Field (+) Euclidean Jordan algebra of + dimension 3 over Algebraic Real Field + + The embeddings work the way you'd expect on the vector + representation of an element:: + + sage: J1 = JordanSpinEJA(3) + sage: J2 = RealSymmetricEJA(2) + sage: J = cartesian_product([J1,J2]) + sage: iota_left = J.cartesian_embedding(0) + sage: iota_right = J.cartesian_embedding(1) + sage: iota_left(J1.zero()) == J.zero() + True + sage: iota_right(J2.zero()) == J.zero() + True + sage: J1.one().to_vector() + (1, 0, 0) + sage: iota_left(J1.one()).to_vector() + (1, 0, 0, 0, 0, 0) + sage: J2.one().to_vector() + (1, 0, 1) + sage: iota_right(J2.one()).to_vector() + (0, 0, 0, 1, 0, 1) + sage: J.one().to_vector() + (1, 0, 0, 1, 0, 1) + + TESTS: + + The answer never changes:: + + sage: set_random_seed() + sage: J1 = random_eja() + sage: J2 = random_eja() + sage: J = cartesian_product([J1,J2]) + sage: E0 = J.cartesian_embedding(0) + sage: E1 = J.cartesian_embedding(0) + sage: E0 == E1 + True + + Composing a projection with the corresponding inclusion should + produce the identity map, and mismatching them should produce + the zero map:: + + sage: set_random_seed() + sage: J1 = random_eja() + sage: J2 = random_eja() + sage: J = cartesian_product([J1,J2]) + sage: iota_left = J.cartesian_embedding(0) + sage: iota_right = J.cartesian_embedding(1) + sage: pi_left = J.cartesian_projection(0) + sage: pi_right = J.cartesian_projection(1) + sage: pi_left*iota_left == J1.one().operator() + True + sage: pi_right*iota_right == J2.one().operator() + True + sage: (pi_left*iota_right).is_zero() + True + sage: (pi_right*iota_left).is_zero() + True + + """ + offset = sum( self.cartesian_factor(k).dimension() + for k in range(i) ) + Ji = self.cartesian_factor(i) + Ei = Ji._module_morphism(lambda j: self.monomial(j + offset), + codomain=self) + return FiniteDimensionalEJAOperator(Ji,self,Ei.matrix()) + + + +FiniteDimensionalEJA.CartesianProduct = CartesianProductEJA + +class RationalBasisCartesianProductEJA(CartesianProductEJA, + RationalBasisEJA): + r""" + A separate class for products of algebras for which we know a + rational basis. SETUP:: - sage: from mjo.eja.eja_algebra import JordanSpinEJA + sage: from mjo.eja.eja_algebra import (HadamardEJA, + ....: JordanSpinEJA, + ....: OctonionHermitianEJA, + ....: RealSymmetricEJA) EXAMPLES: - This multiplication table can be verified by hand:: + This gives us fast characteristic polynomial computations in + product algebras, too:: - sage: J = JordanSpinEJA(4) - sage: e0,e1,e2,e3 = J.gens() - sage: e0*e0 - e0 - sage: e0*e1 - e1 - sage: e0*e2 - e2 - sage: e0*e3 - e3 - sage: e1*e2 - 0 - sage: e1*e3 - 0 - sage: e2*e3 - 0 - """ - @staticmethod - def __classcall_private__(cls, n, field=QQ): - Qs = [] - id_matrix = matrix.identity(field, n) - for i in xrange(n): - ei = id_matrix.column(i) - Qi = matrix.zero(field, n) - Qi.set_row(0, ei) - Qi.set_column(0, ei) - Qi += matrix.diagonal(n, [ei[0]]*n) - # The addition of the diagonal matrix adds an extra ei[0] in the - # upper-left corner of the matrix. - Qi[0,0] = Qi[0,0] * ~field(2) - Qs.append(Qi) - - # The rank of the spin algebra is two, unless we're in a - # one-dimensional ambient space (because the rank is bounded by - # the ambient dimension). - fdeja = super(JordanSpinEJA, cls) - return fdeja.__classcall_private__(cls, field, Qs, rank=min(n,2)) + sage: J1 = JordanSpinEJA(2) + sage: J2 = RealSymmetricEJA(3) + sage: J = cartesian_product([J1,J2]) + sage: J.characteristic_polynomial_of().degree() + 5 + sage: J.rank() + 5 - def inner_product(self, x, y): - return _usual_ip(x,y) + TESTS: + + The ``cartesian_product()`` function only uses the first factor to + decide where the result will live; thus we have to be careful to + check that all factors do indeed have a `_rational_algebra` member + before we try to access it:: + + sage: J1 = OctonionHermitianEJA(1) # no rational basis + sage: J2 = HadamardEJA(2) + sage: cartesian_product([J1,J2]) + Euclidean Jordan algebra of dimension 1 over Algebraic Real Field + (+) Euclidean Jordan algebra of dimension 2 over Algebraic Real Field + sage: cartesian_product([J2,J1]) + Euclidean Jordan algebra of dimension 2 over Algebraic Real Field + (+) Euclidean Jordan algebra of dimension 1 over Algebraic Real Field + + """ + def __init__(self, algebras, **kwargs): + CartesianProductEJA.__init__(self, algebras, **kwargs) + + self._rational_algebra = None + if self.vector_space().base_field() is not QQ: + if all( hasattr(r, "_rational_algebra") for r in algebras ): + self._rational_algebra = cartesian_product([ + r._rational_algebra for r in algebras + ]) + + +RationalBasisEJA.CartesianProduct = RationalBasisCartesianProductEJA + +def random_eja(*args, **kwargs): + J1 = ConcreteEJA.random_instance(*args, **kwargs) + + # This might make Cartesian products appear roughly as often as + # any other ConcreteEJA. + if ZZ.random_element(len(ConcreteEJA.__subclasses__()) + 1) == 0: + # Use random_eja() again so we can get more than two factors. + J2 = random_eja(*args, **kwargs) + J = cartesian_product([J1,J2]) + return J + else: + return J1