r""" 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 a few other example constructions, * :class:`JordanSpinEJA` * :class:`HadamardEJA` * :class:`AlbertEJA` * :class:`TrivialEJA` * :class:`ComplexSkewSymmetricEJA` 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. At a minimum, the following are required to construct a Euclidean Jordan algebra: * A basis of matrices, column vectors, or MatrixAlgebra elements * A Jordan product defined on the basis * Its inner product defined on the basis The real numbers form a Euclidean Jordan algebra when both the Jordan and inner products are the usual multiplication. We use this as our example, and demonstrate a few ways to construct an EJA. First, we can use one-by-one SageMath matrices with algebraic real entries to represent real numbers. We define the Jordan and inner products to be essentially real-number multiplication, with the only difference being that the Jordan product again returns a one-by-one matrix, whereas the inner product must return a scalar. Our basis for the one-by-one matrices is of course the set consisting of a single matrix with its sole entry non-zero:: sage: from mjo.eja.eja_algebra import EJA sage: jp = lambda X,Y: X*Y sage: ip = lambda X,Y: X[0,0]*Y[0,0] sage: b1 = matrix(AA, [[1]]) sage: J1 = EJA((b1,), jp, ip) sage: J1 Euclidean Jordan algebra of dimension 1 over Algebraic Real Field In fact, any positive scalar multiple of that inner-product would work:: sage: ip2 = lambda X,Y: 16*ip(X,Y) sage: J2 = EJA((b1,), jp, ip2) sage: J2 Euclidean Jordan algebra of dimension 1 over Algebraic Real Field But beware that your basis will be orthonormalized _with respect to the given inner-product_ unless you pass ``orthonormalize=False`` to the constructor. For example:: sage: J3 = EJA((b1,), jp, ip2, orthonormalize=False) sage: J3 Euclidean Jordan algebra of dimension 1 over Algebraic Real Field To see the difference, you can take the first and only basis element of the resulting algebra, and ask for it to be converted back into matrix form:: sage: J1.basis()[0].to_matrix() [1] sage: J2.basis()[0].to_matrix() [1/4] sage: J3.basis()[0].to_matrix() [1] Since square roots are used in that process, the default scalar field that we use is the field of algebraic real numbers, ``AA``. You can also Use rational numbers, but only if you either pass ``orthonormalize=False`` or know that orthonormalizing your basis won't stray beyond the rational numbers. The example above would have worked only because ``sqrt(16) == 4`` is rational. Another option for your basis is to use elemebts of a :class:`MatrixAlgebra`:: sage: from mjo.matrix_algebra import MatrixAlgebra sage: A = MatrixAlgebra(1,AA,AA) sage: J4 = EJA(A.gens(), jp, ip) sage: J4 Euclidean Jordan algebra of dimension 1 over Algebraic Real Field sage: J4.basis()[0].to_matrix() +---+ | 1 | +---+ An easier way to view the entire EJA basis in its original (but perhaps orthonormalized) matrix form is to use the ``matrix_basis`` method:: sage: J4.matrix_basis() (+---+ | 1 | +---+,) In particular, a :class:`MatrixAlgebra` is needed to work around the fact that matrices in SageMath must have entries in the same (commutative and associative) ring as its scalars. There are many Euclidean Jordan algebras whose elements are matrices that violate those assumptions. The complex, quaternion, and octonion Hermitian matrices all have entries in a ring (the complex numbers, quaternions, or octonions...) that differs from the algebra's scalar ring (the real numbers). Quaternions are also non-commutative; the octonions are neither commutative nor associative. SETUP:: sage: from mjo.eja.eja_algebra import random_eja EXAMPLES:: sage: random_eja() Euclidean Jordan algebra of dimension... """ from sage.algebras.quatalg.quaternion_algebra import QuaternionAlgebra 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.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 (CartesianProductEJAElement, EJAElement) from mjo.eja.eja_operator import EJAOperator from mjo.eja.eja_utils import _all2list def EuclideanJordanAlgebras(field): r""" The category of Euclidean Jordan algebras over ``field``, which must be a subfield of the real numbers. For now this is just a convenient wrapper around all of the other category axioms that apply to all EJAs. """ category = MagmaticAlgebras(field).FiniteDimensional() category = category.WithBasis().Unital().Commutative() return category class EJA(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. - ``matrix_space`` -- the space that your matrix basis lives in, or ``None`` (the default). So long as your basis does not have length zero you can omit this. But in trivial algebras, it is required. - ``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:: 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:: 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 = EJAElement @staticmethod def _check_input_field(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") @staticmethod def _check_input_axioms(basis, jordan_product, inner_product): 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") def __init__(self, basis, jordan_product, inner_product, field=AA, matrix_space=None, orthonormalize=True, associative=None, check_field=True, check_axioms=True, prefix="b"): n = len(basis) if check_field: self._check_input_field(field) 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. self._check_input_axioms(basis, jordan_product, inner_product) 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) category = EuclideanJordanAlgebras(field) if associative: # Element subalgebras can take advantage of this. category = category.Associative() # 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. 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 hold the orthonormal -> unorthonormal # coordinate transformation. Default to an identity matrix of # the appropriate size to avoid special cases for None # everywhere. self._deortho_matrix = matrix.identity(field,n) 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, as well as the space that its elements live in. # In most cases we can deduce the matrix space, but when # n == 0 (that is, there are no basis elements) we cannot. self._matrix_basis = basis if matrix_space is None: self._matrix_space = self._matrix_basis[0].parent() else: self._matrix_space = matrix_space # 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 ) # Save the span of our matrix basis (when written out as long # vectors) because otherwise we'll have to reconstruct it # every time we want to coerce a matrix into the algebra. self._matrix_span = V.span_of_basis( vector_basis, check=check_axioms) if orthonormalize: # Now "self._matrix_span" is the vector space of our # algebra coordinates. The variables "X0", "X1",... refer # to the entries of vectors in self._matrix_span. Thus to # convert back and forth between the orthonormal # coordinates and the given ones, we need to stick the # original basis in self._matrix_span. U = V.span_of_basis( deortho_vector_basis, check=check_axioms) self._deortho_matrix = matrix.column( 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) zed = self.zero() self._multiplication_table = [ [zed 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 = self._matrix_span.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: 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: 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: Our inner product is "associative," which means the following for a symmetric bilinear form:: 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: 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: J = BilinearFormEJA.random_instance() sage: n = J.dimension() sage: x = J.random_element() sage: y = J.random_element() 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_associative(self): r""" Return whether or not this algebra's Jordan product is associative. SETUP:: sage: from mjo.eja.eja_algebra import ComplexHermitianEJA EXAMPLES:: 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 """ return "Associative" in self.category().axioms() def _is_commutative(self): r""" Whether or not this algebra's multiplication table 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. """ 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. """ 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, ....: 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: The values we've presupplied to the constructors agree with the computation:: sage: J = random_eja() sage: J.is_associative() == J._jordan_product_is_associative() True """ 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 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): """ Construct an element of this algebra or a subalgebra from its EJA element, vector, or matrix representation. This gets called only after the parent element _call_ method fails to find a coercion for the argument. SETUP:: sage: from mjo.eja.eja_algebra import (random_eja, ....: JordanSpinEJA, ....: HadamardEJA, ....: RealSymmetricEJA) 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 Subalgebra elements are embedded into the superalgebra:: sage: J = JordanSpinEJA(3) sage: J.one() b0 sage: x = sum(J.gens()) sage: A = x.subalgebra_generated_by() sage: J(A.one()) b0 TESTS: Ensure that we can convert any element back and forth faithfully between its matrix and algebra representations:: sage: J = random_eja() sage: x = J.random_element() sage: J(x.to_matrix()) == x True 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) if hasattr(elt, 'superalgebra_element'): # Handle subalgebra elements if elt.parent().superalgebra() == self: return elt.superalgebra_element() if hasattr(elt, 'sparse_vector'): # Convert a vector into a column-matrix. We check for # "sparse_vector" and not "column" because matrices also # have a "column" method. elt = elt.column() 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. # # 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. # elt = self._matrix_span.ambient_vector_space()(_all2list(elt)) try: coords = self._matrix_span.coordinate_vector(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_of(self): """ 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 the coordinates of an algebra element: when evaluated at the coordinates of an algebra element with respect to a certain basis, the result is a univariate polynomial (in the one remaining variable ``t``), namely the characteristic polynomial of that element. SETUP:: sage: from mjo.eja.eja_algebra import JordanSpinEJA, TrivialEJA EXAMPLES: The characteristic polynomial in the spin algebra is given in Alizadeh, Example 11.11:: sage: J = JordanSpinEJA(3) sage: p = J.characteristic_polynomial_of(); p X0^2 - X1^2 - X2^2 + (-2*t)*X0 + t^2 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_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. S = PolynomialRing(self.base_ring(),'t') t = S.gen(0) 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:: sage: from mjo.eja.eja_algebra import (HadamardEJA, ....: RealSymmetricEJA) EXAMPLES:: sage: J = HadamardEJA(2) sage: J.coordinate_polynomial_ring() Multivariate Polynomial Ring in X0, X1... sage: J = RealSymmetricEJA(3,field=QQ,orthonormalize=False) sage: J.coordinate_polynomial_ring() Multivariate Polynomial Ring in X0, X1, X2, X3, X4, X5... """ var_names = tuple( "X%d" % z for z in range(self.dimension()) ) return PolynomialRing(self.base_ring(), var_names) 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: Our inner product is "associative," which means the following for a symmetric bilinear form:: 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: 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: J = BilinearFormEJA.random_instance() sage: n = J.dimension() sage: x = J.random_element() sage: y = J.random_element() 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 """ 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 | +----++----+----+----+----+ """ 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) ] return table(M, header_row=True, header_column=True, frame=True) 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:: sage: from mjo.eja.eja_algebra import (JordanSpinEJA, ....: RealSymmetricEJA) EXAMPLES:: sage: J = RealSymmetricEJA(2) sage: J.basis() Finite family {0: b0, 1: b1, 2: b2} sage: J.matrix_basis() ( [1 0] [ 0 0.7071067811865475?] [0 0] [0 0], [0.7071067811865475? 0], [0 1] ) :: sage: J = JordanSpinEJA(2) sage: J.basis() Finite family {0: b0, 1: b1} sage: J.matrix_basis() ( [1] [0] [0], [1] ) """ return self._matrix_basis def matrix_space(self): """ 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). "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 (ComplexHermitianEJA, ....: JordanSpinEJA, ....: QuaternionHermitianEJA, ....: TrivialEJA) EXAMPLES: 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() Module of 2 by 2 matrices with entries in Algebraic Field over the scalar ring 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._matrix_space @cached_method def one(self): """ Return the unit element of this algebra. SETUP:: sage: from mjo.eja.eja_algebra import (HadamardEJA, ....: random_eja) EXAMPLES: We can compute unit element in the Hadamard EJA:: sage: J = HadamardEJA(5) sage: J.one() b0 + b1 + b2 + b3 + b4 The unit element in the Hadamard EJA is inherited in the subalgebras generated by its elements:: 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 TESTS: The identity element acts like the identity, regardless of whether or not we orthonormalize:: 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(orthonormalize=False) sage: y = A.random_element() sage: A.one()*y == y and y*A.one() == y True :: 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 The matrix of the unit element's operator is the identity, regardless of the base field and whether or not we orthonormalize:: 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(orthonormalize=False) sage: actual = A.one().operator().matrix() sage: expected = matrix.identity(A.base_ring(), A.dimension()) sage: actual == expected True :: 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 Ensure that the cached unit element (often precomputed by hand) agrees with the computed one:: sage: J = random_eja() sage: cached = J.one() sage: J.one.clear_cache() sage: J.one() == cached True :: sage: J = random_eja(field=QQ, orthonormalize=False) sage: cached = J.one() sage: J.one.clear_cache() sage: J.one() == cached 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. V = VectorSpace(self.base_ring(), self.dimension()**2) oper_vecs = [ V(g.operator().matrix().list()) for g in self.gens() ] # 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 = V( matrix.identity(self.base_ring(), self.dimension()).list() ) # 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)) def peirce_decomposition(self, c): """ The Peirce decomposition of this algebra relative to the idempotent ``c``. In the future, this can be extended to a complete system of orthogonal idempotents. INPUT: - ``c`` -- an idempotent of this algebra. OUTPUT: A triple (J0, J5, J1) containing two subalgebras and one subspace of this algebra, - ``J0`` -- the algebra on the eigenspace of ``c.operator()`` corresponding to the eigenvalue zero. - ``J5`` -- the eigenspace (NOT a subalgebra) of ``c.operator()`` corresponding to the eigenvalue one-half. - ``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. SETUP:: sage: from mjo.eja.eja_algebra import random_eja, RealSymmetricEJA 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] TESTS: Every algebra decomposes trivially with respect to its identity element:: 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 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: 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 """ 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((), check_axioms=False) 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) return (J0, J5, J1) def random_element(self, thorough=False): r""" Return a random element of this algebra. 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. INPUT: - ``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 """ # For a general base ring... maybe we can trust this to do the # right thing? Unlikely, but. V = self.vector_space() if self.base_ring() is AA and not thorough: # Now that AA generates actually random random elements # (post Trac 30875), we only need to de-thorough the # randomness when asked to. V = V.change_ring(QQ) v = V.random_element() return self.from_vector(V.coordinate_vector(v)) def random_elements(self, count, thorough=False): """ Return ``count`` random elements as a tuple. INPUT: - ``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:: sage: from mjo.eja.eja_algebra import JordanSpinEJA EXAMPLES:: 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 """ return tuple( self.random_element(thorough) for idx in range(count) ) def operator_polynomial_matrix(self): r""" Return the matrix of polynomials (over this algebra's :meth:`coordinate_polynomial_ring`) that, when evaluated at the basis coordinates of an element `x`, produces the basis representation of `L_{x}`. SETUP:: sage: from mjo.eja.eja_algebra import (HadamardEJA, ....: JordanSpinEJA) EXAMPLES:: sage: J = HadamardEJA(4) sage: L_x = J.operator_polynomial_matrix() sage: L_x [X0 0 0 0] [ 0 X1 0 0] [ 0 0 X2 0] [ 0 0 0 X3] sage: x = J.one() sage: d = zip(J.coordinate_polynomial_ring().gens(), x.to_vector()) sage: L_x.subs(dict(d)) [1 0 0 0] [0 1 0 0] [0 0 1 0] [0 0 0 1] :: sage: J = JordanSpinEJA(4) sage: L_x = J.operator_polynomial_matrix() sage: L_x [X0 X1 X2 X3] [X1 X0 0 0] [X2 0 X0 0] [X3 0 0 X0] sage: x = J.one() sage: d = zip(J.coordinate_polynomial_ring().gens(), x.to_vector()) sage: L_x.subs(dict(d)) [1 0 0 0] [0 1 0 0] [0 0 1 0] [0 0 0 1] """ R = self.coordinate_polynomial_ring() 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( v*self.monomial(k).operator().matrix()[i,j] for (k,v) in enumerate(R.gens()) ) n = self.dimension() return matrix(R, n, n, L_x_i_j) @cached_method def _charpoly_coefficients(self): r""" The `r` polynomial coefficients of the "characteristic polynomial of" function. SETUP:: sage: from mjo.eja.eja_algebra import random_eja TESTS: The theory shows that these are all homogeneous polynomials of a known degree:: sage: J = random_eja() sage: all(p.is_homogeneous() for p in J._charpoly_coefficients()) True """ n = self.dimension() R = self.coordinate_polynomial_ring() F = R.fraction_field() L_x = self.operator_polynomial_matrix() 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. 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. SETUP:: sage: from mjo.eja.eja_algebra import (HadamardEJA, ....: JordanSpinEJA, ....: RealSymmetricEJA, ....: ComplexHermitianEJA, ....: QuaternionHermitianEJA, ....: random_eja) EXAMPLES: The rank of the Jordan spin algebra is always two:: sage: JordanSpinEJA(2).rank() 2 sage: JordanSpinEJA(3).rank() 2 sage: JordanSpinEJA(4).rank() 2 The rank of the `n`-by-`n` Hermitian real, complex, or quaternion matrices is `n`:: sage: RealSymmetricEJA(4).rank() 4 sage: ComplexHermitianEJA(3).rank() 3 sage: QuaternionHermitianEJA(2).rank() 2 TESTS: 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:: sage: J = random_eja() sage: r = J.rank() sage: r in ZZ True sage: r > 0 or (r == 0 and J.is_trivial()) True Ensure that computing the rank actually works, since the ranks of all simple algebras are known and will be cached by default:: 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 """ return len(self._charpoly_coefficients()) def subalgebra(self, basis, **kwargs): r""" Create a subalgebra of this algebra from the given basis. """ from mjo.eja.eja_subalgebra import EJASubalgebra return EJASubalgebra(self, basis, **kwargs) def vector_space(self): """ Return the vector space that underlies this algebra. SETUP:: sage: from mjo.eja.eja_algebra import RealSymmetricEJA EXAMPLES:: sage: J = RealSymmetricEJA(2) sage: J.vector_space() Vector space of dimension 3 over... """ return self.zero().to_vector().parent().ambient_vector_space() class RationalBasisEJA(EJA): r""" Algebras whose supplied basis elements have all rational entries. SETUP:: sage: from mjo.eja.eja_algebra import BilinearFormEJA EXAMPLES: The supplied basis is orthonormalized by default:: 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] ) """ 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 = EJA( basis, jordan_product, inner_product, field=QQ, matrix_space=self.matrix_space(), associative=self.is_associative(), orthonormalize=False, check_field=False, check_axioms=False) def rational_algebra(self): # Using None as a flag here (rather than just assigning "self" # to self._rational_algebra by default) feels a little bit # more sane to me in a garbage-collected environment. if self._rational_algebra is None: return self else: return self._rational_algebra @cached_method def _charpoly_coefficients(self): r""" SETUP:: sage: from mjo.eja.eja_algebra import (BilinearFormEJA, ....: JordanSpinEJA) EXAMPLES: 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):: sage: J = JordanSpinEJA(3) sage: J._charpoly_coefficients() (X0^2 - X1^2 - X2^2, -2*X0) 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 self: # Bypass the hijinks if they won't benefit us. return super()._charpoly_coefficients() # Do the computation over the rationals. a = ( a_i.change_ring(self.base_ring()) for a_i in self.rational_algebra()._charpoly_coefficients() ) # Convert our coordinate variables into deorthonormalized ones # and substitute them into the deorthonormalized charpoly # coefficients. 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(EJA): 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." SETUP:: sage: from mjo.eja.eja_algebra import ConcreteEJA TESTS: Our basis is normalized with respect to the algebra's inner product, unless we specify otherwise:: 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 = ConcreteEJA.random_instance() sage: x = J.random_element() sage: x.operator().is_self_adjoint() True """ @staticmethod def _max_random_instance_dimension(): r""" The maximum dimension of any random instance. Ten dimensions seems to be about the point where everything takes a turn for the worse. And dimension ten (but not nine) allows the 4-by-4 real Hermitian matrices, the 2-by-2 quaternion Hermitian matrices, and the 2-by-2 octonion Hermitian matrices. """ return 10 @staticmethod def _max_random_instance_size(max_dimension): """ Return an integer "size" that is an upper bound on the size of this algebra when it is used in a random test case. This size (which can be passed to the algebra's constructor) is itself based on the ``max_dimension`` parameter. This method must be implemented in each subclass. """ raise NotImplementedError @classmethod def random_instance(cls, max_dimension=None, *args, **kwargs): """ Return a random instance of this type of algebra whose dimension is less than or equal to the lesser of ``max_dimension`` and the value returned by ``_max_random_instance_dimension()``. If the dimension bound is omitted, then only the ``_max_random_instance_dimension()`` is used as a bound. This method should be implemented in each subclass. SETUP:: sage: from mjo.eja.eja_algebra import ConcreteEJA TESTS: Both the class bound and the ``max_dimension`` argument are upper bounds on the dimension of the algebra returned:: sage: from sage.misc.prandom import choice sage: eja_class = choice(ConcreteEJA.__subclasses__()) sage: class_max_d = eja_class._max_random_instance_dimension() sage: J = eja_class.random_instance(max_dimension=20, ....: field=QQ, ....: orthonormalize=False) sage: J.dimension() <= class_max_d True sage: J = eja_class.random_instance(max_dimension=2, ....: field=QQ, ....: orthonormalize=False) sage: J.dimension() <= 2 True """ from sage.misc.prandom import choice eja_class = choice(cls.__subclasses__()) # These all bubble up to the RationalBasisEJA superclass # constructor, so any (kw)args valid there are also valid # here. return eja_class.random_instance(max_dimension, *args, **kwargs) class HermitianMatrixEJA(EJA): @staticmethod def _denormalized_basis(A): """ Returns a basis for the given Hermitian matrix space. 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. SETUP:: sage: from mjo.hurwitz import (ComplexMatrixAlgebra, ....: QuaternionMatrixAlgebra, ....: OctonionMatrixAlgebra) sage: from mjo.eja.eja_algebra import HermitianMatrixEJA TESTS:: sage: n = ZZ.random_element(1,5) sage: A = MatrixSpace(QQ, n) sage: B = HermitianMatrixEJA._denormalized_basis(A) sage: all( M.is_hermitian() for M in B) True :: sage: n = ZZ.random_element(1,5) sage: A = ComplexMatrixAlgebra(n, scalars=QQ) sage: B = HermitianMatrixEJA._denormalized_basis(A) sage: all( M.is_hermitian() for M in B) True :: sage: n = ZZ.random_element(1,5) sage: A = QuaternionMatrixAlgebra(n, scalars=QQ) sage: B = HermitianMatrixEJA._denormalized_basis(A) sage: all( M.is_hermitian() for M in B ) True :: sage: n = ZZ.random_element(1,5) sage: A = OctonionMatrixAlgebra(n, scalars=QQ) sage: B = HermitianMatrixEJA._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 ) @staticmethod def jordan_product(X,Y): return (X*Y + Y*X)/2 @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:: sage: from mjo.eja.eja_algebra import (RealSymmetricEJA, ....: ComplexHermitianEJA, ....: QuaternionHermitianEJA, ....: OctonionHermitianEJA) EXAMPLES:: sage: J = RealSymmetricEJA(2,field=QQ,orthonormalize=False) sage: I = J.one().to_matrix() sage: J.trace_inner_product(I, -I) -2 :: sage: J = ComplexHermitianEJA(2,field=QQ,orthonormalize=False) sage: I = J.one().to_matrix() sage: J.trace_inner_product(I, -I) -2 :: sage: J = QuaternionHermitianEJA(2,field=QQ,orthonormalize=False) sage: I = J.one().to_matrix() sage: J.trace_inner_product(I, -I) -2 :: sage: J = OctonionHermitianEJA(2,field=QQ,orthonormalize=False) sage: I = J.one().to_matrix() sage: J.trace_inner_product(I, -I) -2 """ 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] # Works for real and complex numbers. return tr.real() def __init__(self, matrix_space, **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 super().__init__(self._denormalized_basis(matrix_space), self.jordan_product, self.trace_inner_product, field=matrix_space.base_ring(), matrix_space=matrix_space, **kwargs) self.rank.set_cache(matrix_space.nrows()) self.one.set_cache( self(matrix_space.one()) ) class RealSymmetricEJA(HermitianMatrixEJA, RationalBasisEJA, ConcreteEJA): """ 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. SETUP:: sage: from mjo.eja.eja_algebra import RealSymmetricEJA EXAMPLES:: 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: The dimension of this algebra is `(n^2 + n) / 2`:: sage: d = RealSymmetricEJA._max_random_instance_dimension() sage: n = RealSymmetricEJA._max_random_instance_size(d) sage: J = RealSymmetricEJA(n) sage: J.dimension() == (n^2 + n)/2 True The Jordan multiplication is what we think it is:: 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) We can construct the (trivial) algebra of rank zero:: sage: RealSymmetricEJA(0) Euclidean Jordan algebra of dimension 0 over Algebraic Real Field """ @staticmethod def _max_random_instance_size(max_dimension): # Obtained by solving d = (n^2 + n)/2. # The ZZ-int-ZZ thing is just "floor." return ZZ(int(ZZ(8*max_dimension + 1).sqrt()/2 - 1/2)) @classmethod def random_instance(cls, max_dimension=None, *args, **kwargs): """ Return a random instance of this type of algebra. """ class_max_d = cls._max_random_instance_dimension() if (max_dimension is None or max_dimension > class_max_d): max_dimension = class_max_d max_size = cls._max_random_instance_size(max_dimension) n = ZZ.random_element(max_size + 1) return cls(n, **kwargs) def __init__(self, n, field=AA, **kwargs): A = MatrixSpace(field, n) super().__init__(A, **kwargs) from mjo.eja.eja_cache import real_symmetric_eja_coeffs a = real_symmetric_eja_coeffs(self) if a is not None: self.rational_algebra()._charpoly_coefficients.set_cache(a) class ComplexHermitianEJA(HermitianMatrixEJA, RationalBasisEJA, ConcreteEJA): """ 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 ComplexHermitianEJA EXAMPLES: In theory, our "field" can be any subfield of the reals, but we can't use inexact real fields at the moment because SageMath doesn't know how to convert their elements into complex numbers, or even into algebraic reals:: sage: QQbar(RDF(1)) Traceback (most recent call last): ... TypeError: Illegal initializer for algebraic number sage: AA(RR(1)) Traceback (most recent call last): ... TypeError: Illegal initializer for algebraic number TESTS: The dimension of this algebra is `n^2`:: sage: d = ComplexHermitianEJA._max_random_instance_dimension() sage: n = ComplexHermitianEJA._max_random_instance_size(d) sage: J = ComplexHermitianEJA(n) sage: J.dimension() == n^2 True The Jordan multiplication is what we think it is:: 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 """ def __init__(self, n, field=AA, **kwargs): from mjo.hurwitz import ComplexMatrixAlgebra A = ComplexMatrixAlgebra(n, scalars=field) super().__init__(A, **kwargs) from mjo.eja.eja_cache import complex_hermitian_eja_coeffs a = complex_hermitian_eja_coeffs(self) if a is not None: self.rational_algebra()._charpoly_coefficients.set_cache(a) @staticmethod def _max_random_instance_size(max_dimension): # Obtained by solving d = n^2. # The ZZ-int-ZZ thing is just "floor." return ZZ(int(ZZ(max_dimension).sqrt())) @classmethod def random_instance(cls, max_dimension=None, *args, **kwargs): """ Return a random instance of this type of algebra. """ class_max_d = cls._max_random_instance_dimension() if (max_dimension is None or max_dimension > class_max_d): max_dimension = class_max_d max_size = cls._max_random_instance_size(max_dimension) n = ZZ.random_element(max_size + 1) return cls(n, **kwargs) class QuaternionHermitianEJA(HermitianMatrixEJA, RationalBasisEJA, ConcreteEJA): 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 QuaternionHermitianEJA EXAMPLES: 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: The dimension of this algebra is `2*n^2 - n`:: sage: d = QuaternionHermitianEJA._max_random_instance_dimension() sage: n = QuaternionHermitianEJA._max_random_instance_size(d) sage: J = QuaternionHermitianEJA(n) sage: J.dimension() == 2*(n^2) - n True The Jordan multiplication is what we think it is:: 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 We can change the generator prefix:: sage: QuaternionHermitianEJA(2, prefix='a').gens() (a0, a1, a2, a3, a4, a5) We can construct the (trivial) algebra of rank zero:: sage: QuaternionHermitianEJA(0) Euclidean Jordan algebra of dimension 0 over Algebraic Real Field """ def __init__(self, n, field=AA, **kwargs): from mjo.hurwitz import QuaternionMatrixAlgebra A = QuaternionMatrixAlgebra(n, scalars=field) super().__init__(A, **kwargs) from mjo.eja.eja_cache import quaternion_hermitian_eja_coeffs a = quaternion_hermitian_eja_coeffs(self) if a is not None: self.rational_algebra()._charpoly_coefficients.set_cache(a) @staticmethod def _max_random_instance_size(max_dimension): r""" The maximum rank of a random QuaternionHermitianEJA. """ # Obtained by solving d = 2n^2 - n. # The ZZ-int-ZZ thing is just "floor." return ZZ(int(ZZ(8*max_dimension + 1).sqrt()/4 + 1/4)) @classmethod def random_instance(cls, max_dimension=None, *args, **kwargs): """ Return a random instance of this type of algebra. """ class_max_d = cls._max_random_instance_dimension() if (max_dimension is None or max_dimension > class_max_d): max_dimension = class_max_d max_size = cls._max_random_instance_size(max_dimension) n = ZZ.random_element(max_size + 1) return cls(n, **kwargs) class OctonionHermitianEJA(HermitianMatrixEJA, RationalBasisEJA, ConcreteEJA): r""" SETUP:: sage: from mjo.eja.eja_algebra import (EJA, ....: OctonionHermitianEJA) sage: from mjo.hurwitz import Octonions, OctonionMatrixAlgebra EXAMPLES: 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: A = OctonionMatrixAlgebra(2,Octonions(QQ),QQ) sage: b = OctonionHermitianEJA._denormalized_basis(A) 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 = EJA(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: 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 """ @staticmethod def _max_random_instance_size(max_dimension): r""" The maximum rank of a random OctonionHermitianEJA. """ # There's certainly a formula for this, but with only four # cases to worry about, I'm not that motivated to derive it. if max_dimension >= 27: return 3 elif max_dimension >= 10: return 2 elif max_dimension >= 1: return 1 else: return 0 @classmethod def random_instance(cls, max_dimension=None, *args, **kwargs): """ Return a random instance of this type of algebra. """ class_max_d = cls._max_random_instance_dimension() if (max_dimension is None or max_dimension > class_max_d): max_dimension = class_max_d max_size = cls._max_random_instance_size(max_dimension) n = ZZ.random_element(max_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__(A, **kwargs) from mjo.eja.eja_cache import octonion_hermitian_eja_coeffs a = octonion_hermitian_eja_coeffs(self) if a is not None: self.rational_algebra()._charpoly_coefficients.set_cache(a) 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 AlbertEJA EXAMPLES:: 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 """ def __init__(self, *args, **kwargs): super().__init__(3, *args, **kwargs) class HadamardEJA(RationalBasisEJA, ConcreteEJA): """ 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 HadamardEJA EXAMPLES: This multiplication table can be verified by hand:: 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 TESTS: We can change the generator prefix:: sage: HadamardEJA(3, prefix='r').gens() (r0, r1, r2) """ def __init__(self, n, field=AA, **kwargs): MS = MatrixSpace(field, n, 1) if n == 0: jordan_product = lambda x,y: x inner_product = lambda x,y: x else: def jordan_product(x,y): return MS( 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( MS(b) for b in FreeModule(field, n).basis() ) super().__init__(column_basis, jordan_product, inner_product, field=field, matrix_space=MS, associative=True, **kwargs) self.rank.set_cache(n) self.one.set_cache( self.sum(self.gens()) ) @staticmethod def _max_random_instance_dimension(): r""" There's no reason to go higher than five here. That's enough to get the point across. """ return 5 @staticmethod def _max_random_instance_size(max_dimension): r""" The maximum size (=dimension) of a random HadamardEJA. """ return max_dimension @classmethod def random_instance(cls, max_dimension=None, *args, **kwargs): """ Return a random instance of this type of algebra. """ class_max_d = cls._max_random_instance_dimension() if (max_dimension is None or max_dimension > class_max_d): max_dimension = class_max_d max_size = cls._max_random_instance_size(max_dimension) n = ZZ.random_element(max_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 (BilinearFormEJA, ....: JordanSpinEJA) EXAMPLES: 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: We can create a zero-dimensional algebra:: sage: B = matrix.identity(AA,0) sage: J = BilinearFormEJA(B) sage: J.basis() Finite family {} 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: 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 """ 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 n = B.nrows() MS = MatrixSpace(field, n, 1) def inner_product(x,y): return (y.T*B*x)[0,0] def jordan_product(x,y): 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 MS([z0] + zbar.list()) column_basis = tuple( MS(b) 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, matrix_space=MS, 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 _max_random_instance_dimension(): r""" There's no reason to go higher than five here. That's enough to get the point across. """ return 5 @staticmethod def _max_random_instance_size(max_dimension): r""" The maximum size (=dimension) of a random BilinearFormEJA. """ return max_dimension @classmethod def random_instance(cls, max_dimension=None, *args, **kwargs): """ Return a random instance of this algebra. """ class_max_d = cls._max_random_instance_dimension() if (max_dimension is None or max_dimension > class_max_d): max_dimension = class_max_d max_size = cls._max_random_instance_size(max_dimension) n = ZZ.random_element(max_size + 1) if n.is_zero(): B = matrix.identity(ZZ, n) return cls(B, **kwargs) 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 ] ]) return cls(B, **kwargs) class JordanSpinEJA(BilinearFormEJA): """ 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 JordanSpinEJA EXAMPLES: This multiplication table can be verified by hand:: 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 We can change the generator prefix:: sage: JordanSpinEJA(2, prefix='B').gens() (B0, B1) TESTS: Ensure that we have the usual inner product on `R^n`:: 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) @classmethod def random_instance(cls, max_dimension=None, *args, **kwargs): """ Return a random instance of this type of algebra. Needed here to override the implementation for ``BilinearFormEJA``. """ class_max_d = cls._max_random_instance_dimension() if (max_dimension is None or max_dimension > class_max_d): max_dimension = class_max_d max_size = cls._max_random_instance_size(max_dimension) n = ZZ.random_element(max_size + 1) return cls(n, **kwargs) class TrivialEJA(RationalBasisEJA, ConcreteEJA): """ The trivial Euclidean Jordan algebra consisting of only a zero element. SETUP:: sage: from mjo.eja.eja_algebra import TrivialEJA 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, field=AA, **kwargs): jordan_product = lambda x,y: x inner_product = lambda x,y: field.zero() basis = () MS = MatrixSpace(field,0) # 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, field=field, matrix_space=MS, **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, max_dimension=None, *args, **kwargs): # We don't take a "size" argument so the superclass method is # inappropriate for us. The ``max_dimension`` argument is # included so that if this method is called generically with a # ``max_dimension=`` argument, we don't try to pass # it on to the algebra constructor. return cls(**kwargs) class CartesianProductEJA(EJA): 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:: sage: from mjo.eja.eja_algebra import (random_eja, ....: CartesianProductEJA, ....: ComplexHermitianEJA, ....: HadamardEJA, ....: JordanSpinEJA, ....: RealSymmetricEJA) EXAMPLES: The Jordan product is inherited from our factors and implemented by our CombinatorialFreeModule Cartesian product superclass:: 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 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 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 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 | +----++----+----+----+ The "matrix space" of a Cartesian product always consists of ordered pairs (or triples, or...) whose components are the matrix spaces of its factors:: sage: J1 = HadamardEJA(2) sage: J2 = ComplexHermitianEJA(2) sage: J = cartesian_product([J1,J2]) sage: J.matrix_space() The Cartesian product of (Full MatrixSpace of 2 by 1 dense matrices over Algebraic Real Field, Module of 2 by 2 matrices with entries in Algebraic Field over the scalar ring Algebraic Real Field) sage: J.one().to_matrix()[0] [1] [1] sage: J.one().to_matrix()[1] +---+---+ | 1 | 0 | +---+---+ | 0 | 1 | +---+---+ 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: 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 """ Element = CartesianProductEJAElement 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") # Figure out the category to use. associative = all( f.is_associative() for f in factors ) category = EuclideanJordanAlgebras(field) if associative: category = category.Associative() category = category.join([category, category.CartesianProducts()]) # Compute my matrix space. 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. This category isn't perfect, but is good # enough for what we need to do. MS_cat = MagmaticAlgebras(field).FiniteDimensional().WithBasis() MS_cat = MS_cat.Unital().CartesianProducts() MS_factors = tuple( J.matrix_space() for J in factors ) from sage.sets.cartesian_product import CartesianProduct self._matrix_space = CartesianProduct(MS_factors, MS_cat) self._matrix_basis = [] zero = self._matrix_space.zero() for i in range(m): for b in factors[i].matrix_basis(): z = list(zero) z[i] = b self._matrix_basis.append(z) self._matrix_basis = tuple( self._matrix_space(b) for b in self._matrix_basis ) n = len(self._matrix_basis) # We already have what we need for the super-superclass constructor. CombinatorialFreeModule.__init__(self, field, range(n), prefix="b", category=category, bracket=False) # Now create the vector space for the algebra, which will have # its own set of non-ambient coordinates (in terms of the # supplied basis). degree = sum( f._matrix_span.ambient_vector_space().degree() for f in factors ) V = VectorSpace(field, degree) vector_basis = tuple( V(_all2list(b)) for b in self._matrix_basis ) # Save the span of our matrix basis (when written out as long # vectors) because otherwise we'll have to reconstruct it # every time we want to coerce a matrix into the algebra. self._matrix_span = V.span_of_basis( vector_basis, check=False) # Since we don't (re)orthonormalize the basis, the FDEJA # constructor is going to set self._deortho_matrix to the # identity matrix. Here we set it to the correct value using # the deortho matrices from our factors. self._deortho_matrix = matrix.block_diagonal( [J._deortho_matrix for J in factors] ) self._inner_product_matrix = matrix.block_diagonal( [J._inner_product_matrix for J in factors] ) self._inner_product_matrix._cache = {'hermitian': True} self._inner_product_matrix.set_immutable() # Building the multiplication table is a bit more tricky # because we have to embed the entries of the factors' # multiplication tables into the product EJA. zed = self.zero() self._multiplication_table = [ [zed for j in range(i+1)] for i in range(n) ] # Keep track of an offset that tallies the dimensions of all # previous factors. If the second factor is dim=2 and if the # first one is dim=3, then we want to skip the first 3x3 block # when copying the multiplication table for the second factor. offset = 0 for f in range(m): phi_f = self.cartesian_embedding(f) factor_dim = factors[f].dimension() for i in range(factor_dim): for j in range(i+1): f_ij = factors[f]._multiplication_table[i][j] e = phi_f(f_ij) self._multiplication_table[offset+i][offset+j] = e offset += factor_dim self.rank.set_cache(sum(J.rank() for J in factors)) ones = tuple(J.one().to_matrix() for J in factors) self.one.set_cache(self(ones)) def _sets_keys(self): r""" SETUP:: sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA, ....: RealSymmetricEJA) TESTS: The superclass uses ``_sets_keys()`` to implement its ``cartesian_factors()`` method:: sage: J1 = RealSymmetricEJA(2, ....: field=QQ, ....: orthonormalize=False, ....: prefix="a") sage: J2 = ComplexHermitianEJA(2,field=QQ,orthonormalize=False) sage: J = cartesian_product([J1,J2]) sage: x = sum(i*J.gens()[i] for i in range(len(J.gens()))) sage: x.cartesian_factors() (a1 + 2*a2, 3*b0 + 4*b1 + 5*b2 + 6*b3) """ # Copy/pasted from CombinatorialFreeModule_CartesianProduct, # but returning a tuple instead of a list. return tuple(range(len(self.cartesian_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) @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: 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 EJAOperator(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: 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: 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 EJAOperator(Ji,self,Ei.matrix()) def subalgebra(self, basis, **kwargs): r""" Create a subalgebra of this algebra from the given basis. Only overridden to allow us to use a special Cartesian product subalgebra class. SETUP:: sage: from mjo.eja.eja_algebra import (HadamardEJA, ....: QuaternionHermitianEJA) EXAMPLES: Subalgebras of Cartesian product EJAs have a different class than those of non-Cartesian-product EJAs:: sage: J1 = HadamardEJA(2,field=QQ,orthonormalize=False) sage: J2 = QuaternionHermitianEJA(0,field=QQ,orthonormalize=False) sage: J = cartesian_product([J1,J2]) sage: K1 = J1.subalgebra((J1.one(),), orthonormalize=False) sage: K = J.subalgebra((J.one(),), orthonormalize=False) sage: K1.__class__ is K.__class__ False """ from mjo.eja.eja_subalgebra import CartesianProductEJASubalgebra return CartesianProductEJASubalgebra(self, basis, **kwargs) EJA.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 (EJA, ....: HadamardEJA, ....: JordanSpinEJA, ....: RealSymmetricEJA) EXAMPLES: This gives us fast characteristic polynomial computations in product algebras, too:: 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 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()`` method before we construct an algebra that claims to have a rational basis:: sage: J1 = HadamardEJA(2) sage: jp = lambda X,Y: X*Y sage: ip = lambda X,Y: X[0,0]*Y[0,0] sage: b1 = matrix(QQ, [[1]]) sage: J2 = EJA((b1,), jp, ip) sage: cartesian_product([J2,J1]) # factor one not RationalBasisEJA Euclidean Jordan algebra of dimension 1 over Algebraic Real Field (+) Euclidean Jordan algebra of dimension 2 over Algebraic Real Field sage: cartesian_product([J1,J2]) # factor one is RationalBasisEJA Traceback (most recent call last): ... ValueError: factor not a RationalBasisEJA """ def __init__(self, algebras, **kwargs): if not all( hasattr(r, "rational_algebra") for r in algebras ): raise ValueError("factor not a RationalBasisEJA") CartesianProductEJA.__init__(self, algebras, **kwargs) @cached_method def rational_algebra(self): if self.base_ring() is QQ: return self return cartesian_product([ r.rational_algebra() for r in self.cartesian_factors() ]) RationalBasisEJA.CartesianProduct = RationalBasisCartesianProductEJA def random_eja(max_dimension=None, *args, **kwargs): r""" SETUP:: sage: from mjo.eja.eja_algebra import random_eja TESTS:: sage: n = ZZ.random_element(1,5) sage: J = random_eja(max_dimension=n, field=QQ, orthonormalize=False) sage: J.dimension() <= n True """ # Use the ConcreteEJA default as the total upper bound (regardless # of any whether or not any individual factors set a lower limit). if max_dimension is None: max_dimension = ConcreteEJA._max_random_instance_dimension() J1 = ConcreteEJA.random_instance(max_dimension, *args, **kwargs) # Roll the dice to see if we attempt a Cartesian product. dice_roll = ZZ.random_element(len(ConcreteEJA.__subclasses__()) + 1) new_max_dimension = max_dimension - J1.dimension() if new_max_dimension == 0 or dice_roll != 0: # If it's already as big as we're willing to tolerate, just # return it and don't worry about Cartesian products. return J1 else: # Use random_eja() again so we can get more than two factors # if the sub-call also Decides on a cartesian product. J2 = random_eja(new_max_dimension, *args, **kwargs) return cartesian_product([J1,J2]) class ComplexSkewSymmetricEJA(RationalBasisEJA, ConcreteEJA): r""" The skew-symmetric EJA of order `2n` described in Faraut and Koranyi's Exercise III.1.b. It has dimension `2n^2 - n`. It is (not obviously) isomorphic to the QuaternionHermitianEJA of order `n`, as can be inferred by comparing rank/dimension or explicitly from their "characteristic polynomial of" functions, which just so happen to align nicely. SETUP:: sage: from mjo.eja.eja_algebra import (ComplexSkewSymmetricEJA, ....: QuaternionHermitianEJA) sage: from mjo.eja.eja_operator import EJAOperator EXAMPLES: This EJA is isomorphic to the quaternions:: sage: J = ComplexSkewSymmetricEJA(2, field=QQ, orthonormalize=False) sage: K = QuaternionHermitianEJA(2, field=QQ, orthonormalize=False) sage: jordan_isom_matrix = matrix.diagonal(QQ,[-1,1,1,1,1,-1]) sage: phi = EJAOperator(J,K,jordan_isom_matrix) sage: all( phi(x*y) == phi(x)*phi(y) ....: for x in J.gens() ....: for y in J.gens() ) True sage: x,y = J.random_elements(2) sage: phi(x*y) == phi(x)*phi(y) True TESTS: Random elements should satisfy the same conditions that the basis elements do:: sage: K = ComplexSkewSymmetricEJA.random_instance(field=QQ, ....: orthonormalize=False) sage: x,y = K.random_elements(2) sage: z = x*y sage: x = x.to_matrix() sage: y = y.to_matrix() sage: z = z.to_matrix() sage: all( e.is_skew_symmetric() for e in (x,y,z) ) True sage: J = -K.one().to_matrix() sage: all( e*J == J*e.conjugate() for e in (x,y,z) ) True The power law in Faraut & Koranyi's II.7.a is satisfied. We're in a subalgebra of theirs, but powers are still defined the same:: sage: K = ComplexSkewSymmetricEJA.random_instance(field=QQ, ....: orthonormalize=False) sage: x = K.random_element() sage: k = ZZ.random_element(5) sage: actual = x^k sage: J = -K.one().to_matrix() sage: expected = K(-J*(J*x.to_matrix())^k) sage: actual == expected True """ @staticmethod def _max_random_instance_size(max_dimension): # Obtained by solving d = 2n^2 - n, which comes from noticing # that, in 2x2 block form, any element of this algebra has a # free skew-symmetric top-left block, a Hermitian top-right # block, and two bottom blocks that are determined by the top. # The ZZ-int-ZZ thing is just "floor." return ZZ(int(ZZ(8*max_dimension + 1).sqrt()/4 + 1/4)) @classmethod def random_instance(cls, max_dimension=None, *args, **kwargs): """ Return a random instance of this type of algebra. """ class_max_d = cls._max_random_instance_dimension() if (max_dimension is None or max_dimension > class_max_d): max_dimension = class_max_d max_size = cls._max_random_instance_size(max_dimension) n = ZZ.random_element(max_size + 1) return cls(n, **kwargs) @staticmethod def _denormalized_basis(A): """ SETUP:: sage: from mjo.hurwitz import ComplexMatrixAlgebra sage: from mjo.eja.eja_algebra import ComplexSkewSymmetricEJA TESTS: The basis elements are all skew-Hermitian:: sage: d_max = ComplexSkewSymmetricEJA._max_random_instance_dimension() sage: n_max = ComplexSkewSymmetricEJA._max_random_instance_size(d_max) sage: n = ZZ.random_element(n_max + 1) sage: A = ComplexMatrixAlgebra(2*n, scalars=QQ) sage: B = ComplexSkewSymmetricEJA._denormalized_basis(A) sage: all( M.is_skew_symmetric() for M in B) True The basis elements ``b`` all satisfy ``b*J == J*b.conjugate()``, as in the definition of the algebra:: sage: d_max = ComplexSkewSymmetricEJA._max_random_instance_dimension() sage: n_max = ComplexSkewSymmetricEJA._max_random_instance_size(d_max) sage: n = ZZ.random_element(n_max + 1) sage: A = ComplexMatrixAlgebra(2*n, scalars=QQ) sage: I_n = matrix.identity(ZZ, n) sage: J = matrix.block(ZZ, 2, 2, (0, I_n, -I_n, 0), subdivide=False) sage: J = A.from_list(J.rows()) sage: B = ComplexSkewSymmetricEJA._denormalized_basis(A) sage: all( b*J == J*b.conjugate() for b in B ) True """ es = A.entry_algebra_gens() gen = lambda A,m: A.monomial(m) basis = [] # The size of the blocks. We're going to treat these thing as # 2x2 block matrices, # # [ x1 x2 ] # [ -x2-conj x1-conj ] # # where x1 is skew-symmetric and x2 is Hermitian. # m = A.nrows()/2 # We only loop through the top half of the matrix, because the # bottom can be constructed from the top. for i in range(m): # First do the top-left block, which is skew-symmetric. # We can compute the bottom-right block in the process. for j in range(i+1): if i != j: # Skew-symmetry implies zeros for (i == j). for e in es: # Top-left block's entry. E_ij = gen(A, (i,j,e)) E_ij -= gen(A, (j,i,e)) # Bottom-right block's entry. F_ij = gen(A, (i+m,j+m,e)).conjugate() F_ij -= gen(A, (j+m,i+m,e)).conjugate() basis.append(E_ij + F_ij) # Now do the top-right block, which is Hermitian, and compute # the bottom-left block along the way. for j in range(m,i+m+1): if (i+m) == j: # Hermitian matrices have real diagonal entries. # Top-right block's entry. E_ii = gen(A, (i,j,es[0])) # Bottom-left block's entry. Don't conjugate # 'cause it's real. E_ii -= gen(A, (i+m,j-m,es[0])) basis.append(E_ii) else: for e in es: # Top-right block's entry. BEWARE! We're not # reflecting across the main diagonal as in # (i,j)~(j,i). We're only reflecting across # the diagonal for the top-right block. E_ij = gen(A, (i,j,e)) # Shift it back to non-offset coords, transpose, # conjugate, and put it back: # # (i,j) -> (i,j-m) -> (j-m, i) -> (j-m, i+m) E_ij += gen(A, (j-m,i+m,e)).conjugate() # Bottom-left's block's below-diagonal entry. # Just shift the top-right coords down m and # left m. F_ij = -gen(A, (i+m,j-m,e)).conjugate() F_ij += -gen(A, (j,i,e)) # double-conjugate cancels basis.append(E_ij + F_ij) return tuple( basis ) @staticmethod @cached_method def _J_matrix(matrix_space): n = matrix_space.nrows() // 2 F = matrix_space.base_ring() I_n = matrix.identity(F, n) J = matrix.block(F, 2, 2, (0, I_n, -I_n, 0), subdivide=False) return matrix_space.from_list(J.rows()) def J_matrix(self): return ComplexSkewSymmetricEJA._J_matrix(self.matrix_space()) def __init__(self, n, field=AA, **kwargs): # New code; always check the axioms. #if "check_axioms" not in kwargs: kwargs["check_axioms"] = False from mjo.hurwitz import ComplexMatrixAlgebra A = ComplexMatrixAlgebra(2*n, scalars=field) J = ComplexSkewSymmetricEJA._J_matrix(A) def jordan_product(X,Y): return (X*J*Y + Y*J*X)/2 def inner_product(X,Y): return (X.conjugate_transpose()*Y).trace().real() super().__init__(self._denormalized_basis(A), jordan_product, inner_product, field=field, matrix_space=A, **kwargs) # This algebra is conjectured (by me) to be isomorphic to # the quaternion Hermitian EJA of size n, and the rank # would follow from that. #self.rank.set_cache(n) self.one.set_cache( self(-J) )