""" Euclidean Jordan Algebras. These are formally-real Jordan Algebras; specifically those where u^2 + v^2 = 0 implies that u = v = 0. They are used in optimization, and have some additional nice methods beyond what can be supported in a general Jordan Algebra. """ from sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra import FiniteDimensionalAlgebra from sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra_element import FiniteDimensionalAlgebraElement from sage.algebras.quatalg.quaternion_algebra import QuaternionAlgebra from sage.categories.finite_dimensional_algebras_with_basis import FiniteDimensionalAlgebrasWithBasis from sage.functions.other import sqrt from sage.matrix.constructor import matrix from sage.misc.cachefunc import cached_method from sage.misc.prandom import choice from sage.modules.free_module import VectorSpace from sage.modules.free_module_element import vector from sage.rings.integer_ring import ZZ from sage.rings.number_field.number_field import QuadraticField from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing from sage.rings.rational_field import QQ from sage.structure.element import is_Matrix from sage.structure.category_object import normalize_names from mjo.eja.eja_operator import FiniteDimensionalEuclideanJordanAlgebraOperator class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): @staticmethod def __classcall_private__(cls, field, mult_table, names='e', assume_associative=False, category=None, rank=None, natural_basis=None): n = len(mult_table) mult_table = [b.base_extend(field) for b in mult_table] for b in mult_table: b.set_immutable() if not (is_Matrix(b) and b.dimensions() == (n, n)): raise ValueError("input is not a multiplication table") mult_table = tuple(mult_table) cat = FiniteDimensionalAlgebrasWithBasis(field) cat.or_subcategory(category) if assume_associative: cat = cat.Associative() names = normalize_names(n, names) fda = super(FiniteDimensionalEuclideanJordanAlgebra, cls) return fda.__classcall__(cls, field, mult_table, assume_associative=assume_associative, names=names, category=cat, rank=rank, natural_basis=natural_basis) def __init__(self, field, mult_table, names='e', assume_associative=False, category=None, rank=None, natural_basis=None): """ SETUP:: sage: from mjo.eja.eja_algebra import random_eja EXAMPLES: By definition, Jordan multiplication commutes:: sage: set_random_seed() sage: J = random_eja() sage: x = J.random_element() sage: y = J.random_element() sage: x*y == y*x True """ self._rank = rank self._natural_basis = natural_basis self._multiplication_table = mult_table fda = super(FiniteDimensionalEuclideanJordanAlgebra, self) fda.__init__(field, mult_table, names=names, category=category) 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=QQ) Euclidean Jordan algebra of degree 2 over Rational Field sage: JordanSpinEJA(3, field=RDF) Euclidean Jordan algebra of degree 3 over Real Double Field """ fmt = "Euclidean Jordan algebra of degree {} over {}" return fmt.format(self.degree(), self.base_ring()) def _a_regular_element(self): """ Guess a regular element. Needed to compute the basis for our characteristic polynomial coefficients. SETUP:: sage: from mjo.eja.eja_algebra import random_eja TESTS: Ensure that this hacky method succeeds for every algebra that we know how to construct:: sage: set_random_seed() sage: J = random_eja() sage: J._a_regular_element().is_regular() True """ gs = self.gens() z = self.sum( (i+1)*gs[i] for i in range(len(gs)) ) if not z.is_regular(): raise ValueError("don't know a regular element") return z @cached_method def _charpoly_basis_space(self): """ Return the vector space spanned by the basis used in our characteristic polynomial coefficients. This is used not only to compute those coefficients, but also any time we need to evaluate the coefficients (like when we compute the trace or determinant). """ z = self._a_regular_element() V = self.vector_space() V1 = V.span_of_basis( (z**k).vector() for k in range(self.rank()) ) b = (V1.basis() + V1.complement().basis()) return V.span_of_basis(b) @cached_method def _charpoly_coeff(self, i): """ Return the coefficient polynomial "a_{i}" of this algebra's general characteristic polynomial. Having this be a separate cached method lets us compute and store the trace/determinant (a_{r-1} and a_{0} respectively) separate from the entire characteristic polynomial. """ (A_of_x, x, xr, detA) = self._charpoly_matrix_system() R = A_of_x.base_ring() if i >= self.rank(): # Guaranteed by theory return R.zero() # Danger: the in-place modification is done for performance # reasons (reconstructing a matrix with huge polynomial # entries is slow), but I don't know how cached_method works, # so it's highly possible that we're modifying some global # list variable by reference, here. In other words, you # probably shouldn't call this method twice on the same # algebra, at the same time, in two threads Ai_orig = A_of_x.column(i) A_of_x.set_column(i,xr) numerator = A_of_x.det() A_of_x.set_column(i,Ai_orig) # We're relying on the theory here to ensure that each a_i is # indeed back in R, and the added negative signs are to make # the whole charpoly expression sum to zero. return R(-numerator/detA) @cached_method def _charpoly_matrix_system(self): """ Compute the matrix whose entries A_ij are polynomials in X1,...,XN, the vector ``x`` of variables X1,...,XN, the vector corresponding to `x^r` and the determinent of the matrix A = [A_ij]. In other words, all of the fixed (cachable) data needed to compute the coefficients of the characteristic polynomial. """ r = self.rank() n = self.dimension() # Construct a new algebra over a multivariate polynomial ring... names = ['X' + str(i) for i in range(1,n+1)] R = PolynomialRing(self.base_ring(), names) J = FiniteDimensionalEuclideanJordanAlgebra(R, self._multiplication_table, rank=r) idmat = matrix.identity(J.base_ring(), n) W = self._charpoly_basis_space() W = W.change_ring(R.fraction_field()) # Starting with the standard coordinates x = (X1,X2,...,Xn) # and then converting the entries to W-coordinates allows us # to pass in the standard coordinates to the charpoly and get # back the right answer. Specifically, with x = (X1,X2,...,Xn), # we have # # W.coordinates(x^2) eval'd at (standard z-coords) # = # W-coords of (z^2) # = # W-coords of (standard coords of x^2 eval'd at std-coords of z) # # We want the middle equivalent thing in our matrix, but use # the first equivalent thing instead so that we can pass in # standard coordinates. x = J(W(R.gens())) l1 = [matrix.column(W.coordinates((x**k).vector())) for k in range(r)] l2 = [idmat.column(k-1).column() for k in range(r+1, n+1)] A_of_x = matrix.block(R, 1, n, (l1 + l2)) xr = W.coordinates((x**r).vector()) return (A_of_x, x, xr, A_of_x.det()) @cached_method def characteristic_polynomial(self): """ Return a characteristic polynomial that works for all elements of this algebra. 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 EXAMPLES: The characteristic polynomial in the spin algebra is given in Alizadeh, Example 11.11:: sage: J = JordanSpinEJA(3) sage: p = J.characteristic_polynomial(); p X1^2 - X2^2 - X3^2 + (-2*t)*X1 + t^2 sage: xvec = J.one().vector() sage: p(*xvec) t^2 - 2*t + 1 """ r = self.rank() n = self.dimension() # The list of coefficient polynomials a_1, a_2, ..., a_n. a = [ self._charpoly_coeff(i) for i in range(n) ] # We go to a bit of trouble here to reorder the # indeterminates, so that it's easier to evaluate the # characteristic polynomial at x's coordinates and get back # something in terms of t, which is what we want. R = a[0].parent() S = PolynomialRing(self.base_ring(),'t') t = S.gen(0) S = PolynomialRing(S, R.variable_names()) t = S(t) # Note: all entries past the rth should be zero. The # coefficient of the highest power (x^r) is 1, but it doesn't # appear in the solution vector which contains coefficients # for the other powers (to make them sum to x^r). if (r < n): a[r] = 1 # corresponds to x^r else: # When the rank is equal to the dimension, trying to # assign a[r] goes out-of-bounds. a.append(1) # corresponds to x^r return sum( a[k]*(t**k) for k in range(len(a)) ) 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 EXAMPLES: The inner product must satisfy its axiom for this algebra to truly be a Euclidean Jordan Algebra:: sage: set_random_seed() sage: J = random_eja() sage: x = J.random_element() sage: y = J.random_element() sage: z = J.random_element() sage: (x*y).inner_product(z) == y.inner_product(x*z) True """ if (not x in self) or (not y in self): raise TypeError("arguments must live in this algebra") return x.trace_inner_product(y) def natural_basis(self): """ Return a more-natural representation of this algebra's basis. Every finite-dimensional Euclidean Jordan Algebra is a direct sum of five simple algebras, four of which comprise Hermitian matrices. This method returns the original "natural" basis for our underlying vector space. (Typically, the natural basis is used to construct the multiplication table in the first place.) Note that this will always return a matrix. The standard basis in `R^n` will be returned as `n`-by-`1` column matrices. SETUP:: sage: from mjo.eja.eja_algebra import (JordanSpinEJA, ....: RealSymmetricEJA) EXAMPLES:: sage: J = RealSymmetricEJA(2) sage: J.basis() Family (e0, e1, e2) sage: J.natural_basis() ( [1 0] [0 1] [0 0] [0 0], [1 0], [0 1] ) :: sage: J = JordanSpinEJA(2) sage: J.basis() Family (e0, e1) sage: J.natural_basis() ( [1] [0] [0], [1] ) """ if self._natural_basis is None: return tuple( b.vector().column() for b in self.basis() ) else: return self._natural_basis def rank(self): """ Return the rank of this EJA. """ if self._rank is None: raise ValueError("no rank specified at genesis") else: return self._rank 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 Rational Field """ return self.zero().vector().parent().ambient_vector_space() class Element(FiniteDimensionalAlgebraElement): """ An element of a Euclidean Jordan algebra. """ def __dir__(self): """ Oh man, I should not be doing this. This hides the "disabled" methods ``left_matrix`` and ``matrix`` from introspection; in particular it removes them from tab-completion. """ return filter(lambda s: s not in ['left_matrix', 'matrix'], dir(self.__class__) ) def __init__(self, A, elt=None): """ SETUP:: sage: from mjo.eja.eja_algebra import 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): ... ArithmeticError: vector is not in free module """ # Goal: if we're given a matrix, and if it lives in our # parent algebra's "natural ambient space," convert it # into an algebra element. # # The catch is, we make a recursive call after converting # the given matrix into a vector that lives in the algebra. # This we need to try the parent class initializer first, # to avoid recursing forever if we're given something that # already fits into the algebra, but also happens to live # in the parent's "natural ambient space" (this happens with # vectors in R^n). try: FiniteDimensionalAlgebraElement.__init__(self, A, elt) except ValueError: natural_basis = A.natural_basis() if elt in natural_basis[0].matrix_space(): # Thanks for nothing! Matrix spaces aren't vector # spaces in Sage, so we have to figure out its # natural-basis coordinates ourselves. V = VectorSpace(elt.base_ring(), elt.nrows()**2) W = V.span( _mat2vec(s) for s in natural_basis ) coords = W.coordinates(_mat2vec(elt)) FiniteDimensionalAlgebraElement.__init__(self, A, coords) def __pow__(self, n): """ Return ``self`` raised to the power ``n``. Jordan algebras are always power-associative; see for example Faraut and Koranyi, Proposition II.1.2 (ii). .. WARNING: We have to override this because our superclass uses row vectors instead of column vectors! We, on the other hand, assume column vectors everywhere. SETUP:: sage: from mjo.eja.eja_algebra import random_eja EXAMPLES:: sage: set_random_seed() sage: x = random_eja().random_element() sage: x.operator()(x) == (x^2) True A few examples of power-associativity:: sage: set_random_seed() sage: x = random_eja().random_element() sage: x*(x*x)*(x*x) == x^5 True sage: (x*x)*(x*x*x) == x^5 True We also know that powers operator-commute (Koecher, Chapter III, Corollary 1):: sage: set_random_seed() sage: x = random_eja().random_element() sage: m = ZZ.random_element(0,10) sage: n = ZZ.random_element(0,10) sage: Lxm = (x^m).operator() sage: Lxn = (x^n).operator() sage: Lxm*Lxn == Lxn*Lxm True """ if n == 0: return self.parent().one() elif n == 1: return self else: return (self.operator()**(n-1))(self) def apply_univariate_polynomial(self, p): """ Apply the univariate polynomial ``p`` to this element. A priori, SageMath won't allow us to apply a univariate polynomial to an element of an EJA, because we don't know that EJAs are rings (they are usually not associative). Of course, we know that EJAs are power-associative, so the operation is ultimately kosher. This function sidesteps the CAS to get the answer we want and expect. SETUP:: sage: from mjo.eja.eja_algebra import (RealCartesianProductEJA, ....: random_eja) EXAMPLES:: sage: R = PolynomialRing(QQ, 't') sage: t = R.gen(0) sage: p = t^4 - t^3 + 5*t - 2 sage: J = RealCartesianProductEJA(5) sage: J.one().apply_univariate_polynomial(p) == 3*J.one() True TESTS: We should always get back an element of the algebra:: sage: set_random_seed() sage: p = PolynomialRing(QQ, 't').random_element() sage: J = random_eja() sage: x = J.random_element() sage: x.apply_univariate_polynomial(p) in J True """ if len(p.variables()) > 1: raise ValueError("not a univariate polynomial") P = self.parent() R = P.base_ring() # Convert the coeficcients to the parent's base ring, # because a priori they might live in an (unnecessarily) # larger ring for which P.sum() would fail below. cs = [ R(c) for c in p.coefficients(sparse=False) ] return P.sum( cs[k]*(self**k) for k in range(len(cs)) ) def characteristic_polynomial(self): """ Return the characteristic polynomial of this element. SETUP:: sage: from mjo.eja.eja_algebra import RealCartesianProductEJA EXAMPLES: The rank of `R^3` is three, and the minimal polynomial of the identity element is `(t-1)` from which it follows that the characteristic polynomial should be `(t-1)^3`:: sage: J = RealCartesianProductEJA(3) sage: J.one().characteristic_polynomial() t^3 - 3*t^2 + 3*t - 1 Likewise, the characteristic of the zero element in the rank-three algebra `R^{n}` should be `t^{3}`:: sage: J = RealCartesianProductEJA(3) sage: J.zero().characteristic_polynomial() t^3 The characteristic polynomial of an element should evaluate to zero on that element:: sage: set_random_seed() sage: x = RealCartesianProductEJA(3).random_element() sage: p = x.characteristic_polynomial() sage: x.apply_univariate_polynomial(p) 0 """ p = self.parent().characteristic_polynomial() return p(*self.vector()) def inner_product(self, other): """ Return the parent algebra's inner product of myself and ``other``. SETUP:: sage: from mjo.eja.eja_algebra import ( ....: ComplexHermitianEJA, ....: JordanSpinEJA, ....: QuaternionHermitianEJA, ....: RealSymmetricEJA, ....: random_eja) EXAMPLES: The inner product in the Jordan spin algebra is the usual inner product on `R^n` (this example only works because the basis for the Jordan algebra is the standard basis in `R^n`):: sage: J = JordanSpinEJA(3) sage: x = vector(QQ,[1,2,3]) sage: y = vector(QQ,[4,5,6]) sage: x.inner_product(y) 32 sage: J(x).inner_product(J(y)) 32 The inner product on `S^n` is ` = trace(X*Y)`, where multiplication is the usual matrix multiplication in `S^n`, so the inner product of the identity matrix with itself should be the `n`:: sage: J = RealSymmetricEJA(3) sage: J.one().inner_product(J.one()) 3 Likewise, the inner product on `C^n` is ` = Re(trace(X*Y))`, where we must necessarily take the real part because the product of Hermitian matrices may not be Hermitian:: sage: J = ComplexHermitianEJA(3) sage: J.one().inner_product(J.one()) 3 Ditto for the quaternions:: sage: J = QuaternionHermitianEJA(3) sage: J.one().inner_product(J.one()) 3 TESTS: Ensure that we can always compute an inner product, and that it gives us back a real number:: sage: set_random_seed() sage: J = random_eja() sage: x = J.random_element() sage: y = J.random_element() sage: x.inner_product(y) in RR True """ P = self.parent() if not other in P: raise TypeError("'other' must live in the same algebra") return P.inner_product(self, other) def operator_commutes_with(self, other): """ Return whether or not this element operator-commutes with ``other``. SETUP:: sage: from mjo.eja.eja_algebra import random_eja EXAMPLES: The definition of a Jordan algebra says that any element operator-commutes with its square:: sage: set_random_seed() sage: x = random_eja().random_element() sage: x.operator_commutes_with(x^2) True TESTS: Test Lemma 1 from Chapter III of Koecher:: sage: set_random_seed() sage: J = random_eja() sage: u = J.random_element() sage: v = J.random_element() sage: lhs = u.operator_commutes_with(u*v) sage: rhs = v.operator_commutes_with(u^2) sage: lhs == rhs True Test the first polarization identity from my notes, Koecher Chapter III, or from Baes (2.3):: sage: set_random_seed() sage: J = random_eja() sage: x = J.random_element() sage: y = J.random_element() sage: Lx = x.operator() sage: Ly = y.operator() sage: Lxx = (x*x).operator() sage: Lxy = (x*y).operator() sage: bool(2*Lx*Lxy + Ly*Lxx == 2*Lxy*Lx + Lxx*Ly) True Test the second polarization identity from my notes or from Baes (2.4):: sage: set_random_seed() sage: J = random_eja() sage: x = J.random_element() sage: y = J.random_element() sage: z = J.random_element() sage: Lx = x.operator() sage: Ly = y.operator() sage: Lz = z.operator() sage: Lzy = (z*y).operator() sage: Lxy = (x*y).operator() sage: Lxz = (x*z).operator() sage: bool(Lx*Lzy + Lz*Lxy + Ly*Lxz == Lzy*Lx + Lxy*Lz + Lxz*Ly) True Test the third polarization identity from my notes or from Baes (2.5):: sage: set_random_seed() sage: J = random_eja() sage: u = J.random_element() sage: y = J.random_element() sage: z = J.random_element() sage: Lu = u.operator() sage: Ly = y.operator() sage: Lz = z.operator() sage: Lzy = (z*y).operator() sage: Luy = (u*y).operator() sage: Luz = (u*z).operator() sage: Luyz = (u*(y*z)).operator() sage: lhs = Lu*Lzy + Lz*Luy + Ly*Luz sage: rhs = Luyz + Ly*Lu*Lz + Lz*Lu*Ly sage: bool(lhs == rhs) True """ if not other in self.parent(): raise TypeError("'other' must live in the same algebra") A = self.operator() B = other.operator() return (A*B == B*A) def det(self): """ Return my determinant, the product of my eigenvalues. SETUP:: sage: from mjo.eja.eja_algebra import (JordanSpinEJA, ....: random_eja) EXAMPLES:: sage: J = JordanSpinEJA(2) sage: e0,e1 = J.gens() sage: x = sum( J.gens() ) sage: x.det() 0 :: sage: J = JordanSpinEJA(3) sage: e0,e1,e2 = J.gens() sage: x = sum( J.gens() ) sage: x.det() -1 TESTS: An element is invertible if and only if its determinant is non-zero:: sage: set_random_seed() sage: x = random_eja().random_element() sage: x.is_invertible() == (x.det() != 0) True """ P = self.parent() r = P.rank() p = P._charpoly_coeff(0) # The _charpoly_coeff function already adds the factor of # -1 to ensure that _charpoly_coeff(0) is really what # appears in front of t^{0} in the charpoly. However, # we want (-1)^r times THAT for the determinant. return ((-1)**r)*p(*self.vector()) def inverse(self): """ Return the Jordan-multiplicative inverse of this element. ALGORITHM: We appeal to the quadratic representation as in Koecher's Theorem 12 in Chapter III, Section 5. SETUP:: sage: from mjo.eja.eja_algebra import (JordanSpinEJA, ....: random_eja) EXAMPLES: The inverse in the spin factor algebra is given in Alizadeh's Example 11.11:: sage: set_random_seed() sage: n = ZZ.random_element(1,10) sage: J = JordanSpinEJA(n) sage: x = J.random_element() sage: while not x.is_invertible(): ....: x = J.random_element() sage: x_vec = x.vector() sage: x0 = x_vec[0] sage: x_bar = x_vec[1:] sage: coeff = ~(x0^2 - x_bar.inner_product(x_bar)) sage: inv_vec = x_vec.parent()([x0] + (-x_bar).list()) sage: x_inverse = coeff*inv_vec sage: x.inverse() == J(x_inverse) True TESTS: The identity element is its own inverse:: sage: set_random_seed() sage: J = random_eja() sage: J.one().inverse() == J.one() True If an element has an inverse, it acts like one:: sage: set_random_seed() sage: J = random_eja() sage: x = J.random_element() sage: (not x.is_invertible()) or (x.inverse()*x == J.one()) True The inverse of the inverse is what we started with:: sage: set_random_seed() sage: J = random_eja() sage: x = J.random_element() sage: (not x.is_invertible()) or (x.inverse().inverse() == x) True The zero element is never invertible:: sage: set_random_seed() sage: J = random_eja().zero().inverse() Traceback (most recent call last): ... ValueError: element is not invertible """ if not self.is_invertible(): raise ValueError("element is not invertible") return (~self.quadratic_representation())(self) def is_invertible(self): """ Return whether or not this element is invertible. We can't use the superclass method because it relies on the algebra being associative. ALGORITHM: The usual way to do this is to check if the determinant is zero, but we need the characteristic polynomial for the determinant. The minimal polynomial is a lot easier to get, so we use Corollary 2 in Chapter V of Koecher to check whether or not the paren't algebra's zero element is a root of this element's minimal polynomial. SETUP:: sage: from mjo.eja.eja_algebra import random_eja TESTS: The identity element is always invertible:: sage: set_random_seed() sage: J = random_eja() sage: J.one().is_invertible() True The zero element is never invertible:: sage: set_random_seed() sage: J = random_eja() sage: J.zero().is_invertible() False """ zero = self.parent().zero() p = self.minimal_polynomial() return not (p(zero) == zero) def is_nilpotent(self): """ Return whether or not some power of this element is zero. The superclass method won't work unless we're in an associative algebra, and we aren't. However, we generate an assocoative subalgebra and we're nilpotent there if and only if we're nilpotent here (probably). SETUP:: sage: from mjo.eja.eja_algebra import random_eja TESTS: The identity element is never nilpotent:: sage: set_random_seed() sage: random_eja().one().is_nilpotent() False The additive identity is always nilpotent:: sage: set_random_seed() sage: random_eja().zero().is_nilpotent() True """ # The element we're going to call "is_nilpotent()" on. # Either myself, interpreted as an element of a finite- # dimensional algebra, or an element of an associative # subalgebra. elt = None if self.parent().is_associative(): elt = FiniteDimensionalAlgebraElement(self.parent(), self) else: V = self.span_of_powers() assoc_subalg = self.subalgebra_generated_by() # Mis-design warning: the basis used for span_of_powers() # and subalgebra_generated_by() must be the same, and in # the same order! elt = assoc_subalg(V.coordinates(self.vector())) # Recursive call, but should work since elt lives in an # associative algebra. return elt.is_nilpotent() def is_regular(self): """ Return whether or not this is a regular element. SETUP:: sage: from mjo.eja.eja_algebra import JordanSpinEJA EXAMPLES: The identity element always has degree one, but any element linearly-independent from it is regular:: sage: J = JordanSpinEJA(5) sage: J.one().is_regular() False sage: e0, e1, e2, e3, e4 = J.gens() # e0 is the identity sage: for x in J.gens(): ....: (J.one() + x).is_regular() False True True True True """ return self.degree() == self.parent().rank() def degree(self): """ Compute the degree of this element the straightforward way according to the definition; by appending powers to a list and figuring out its dimension (that is, whether or not they're linearly dependent). SETUP:: sage: from mjo.eja.eja_algebra import JordanSpinEJA EXAMPLES:: sage: J = JordanSpinEJA(4) sage: J.one().degree() 1 sage: e0,e1,e2,e3 = J.gens() sage: (e0 - e1).degree() 2 In the spin factor algebra (of rank two), all elements that aren't multiples of the identity are regular:: sage: set_random_seed() sage: n = ZZ.random_element(1,10) sage: J = JordanSpinEJA(n) sage: x = J.random_element() sage: x == x.coefficient(0)*J.one() or x.degree() == 2 True """ return self.span_of_powers().dimension() def left_matrix(self): """ Our parent class defines ``left_matrix`` and ``matrix`` methods whose names are misleading. We don't want them. """ raise NotImplementedError("use operator().matrix() instead") matrix = left_matrix def minimal_polynomial(self): """ Return the minimal polynomial of this element, as a function of the variable `t`. ALGORITHM: We restrict ourselves to the associative subalgebra generated by this element, and then return the minimal polynomial of this element's operator matrix (in that subalgebra). This works by Baes Proposition 2.3.16. SETUP:: sage: from mjo.eja.eja_algebra import (JordanSpinEJA, ....: random_eja) TESTS: The minimal polynomial of the identity and zero elements are always the same:: sage: set_random_seed() sage: J = random_eja() sage: J.one().minimal_polynomial() t - 1 sage: J.zero().minimal_polynomial() t The degree of an element is (by one definition) the degree of its minimal polynomial:: sage: set_random_seed() sage: x = random_eja().random_element() sage: x.degree() == x.minimal_polynomial().degree() True The minimal polynomial and the characteristic polynomial coincide and are known (see Alizadeh, Example 11.11) for all elements of the spin factor algebra that aren't scalar multiples of the identity:: sage: set_random_seed() sage: n = ZZ.random_element(2,10) sage: J = JordanSpinEJA(n) sage: y = J.random_element() sage: while y == y.coefficient(0)*J.one(): ....: y = J.random_element() sage: y0 = y.vector()[0] sage: y_bar = y.vector()[1:] sage: actual = y.minimal_polynomial() sage: t = PolynomialRing(J.base_ring(),'t').gen(0) sage: expected = t^2 - 2*y0*t + (y0^2 - norm(y_bar)^2) sage: bool(actual == expected) True The minimal polynomial should always kill its element:: sage: set_random_seed() sage: x = random_eja().random_element() sage: p = x.minimal_polynomial() sage: x.apply_univariate_polynomial(p) 0 """ V = self.span_of_powers() assoc_subalg = self.subalgebra_generated_by() # Mis-design warning: the basis used for span_of_powers() # and subalgebra_generated_by() must be the same, and in # the same order! elt = assoc_subalg(V.coordinates(self.vector())) return elt.operator().minimal_polynomial() def natural_representation(self): """ Return a more-natural representation of this element. Every finite-dimensional Euclidean Jordan Algebra is a direct sum of five simple algebras, four of which comprise Hermitian matrices. This method returns the original "natural" representation of this element as a Hermitian matrix, if it has one. If not, you get the usual representation. SETUP:: sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA, ....: QuaternionHermitianEJA) EXAMPLES:: sage: J = ComplexHermitianEJA(3) sage: J.one() e0 + e5 + e8 sage: J.one().natural_representation() [1 0 0 0 0 0] [0 1 0 0 0 0] [0 0 1 0 0 0] [0 0 0 1 0 0] [0 0 0 0 1 0] [0 0 0 0 0 1] :: sage: J = QuaternionHermitianEJA(3) sage: J.one() e0 + e9 + e14 sage: J.one().natural_representation() [1 0 0 0 0 0 0 0 0 0 0 0] [0 1 0 0 0 0 0 0 0 0 0 0] [0 0 1 0 0 0 0 0 0 0 0 0] [0 0 0 1 0 0 0 0 0 0 0 0] [0 0 0 0 1 0 0 0 0 0 0 0] [0 0 0 0 0 1 0 0 0 0 0 0] [0 0 0 0 0 0 1 0 0 0 0 0] [0 0 0 0 0 0 0 1 0 0 0 0] [0 0 0 0 0 0 0 0 1 0 0 0] [0 0 0 0 0 0 0 0 0 1 0 0] [0 0 0 0 0 0 0 0 0 0 1 0] [0 0 0 0 0 0 0 0 0 0 0 1] """ B = self.parent().natural_basis() W = B[0].matrix_space() return W.linear_combination(zip(self.vector(), B)) def operator(self): """ Return the left-multiplication-by-this-element operator on the ambient algebra. SETUP:: sage: from mjo.eja.eja_algebra import random_eja TESTS:: sage: set_random_seed() sage: J = random_eja() sage: x = J.random_element() sage: y = J.random_element() sage: x.operator()(y) == x*y True sage: y.operator()(x) == x*y True """ P = self.parent() fda_elt = FiniteDimensionalAlgebraElement(P, self) return FiniteDimensionalEuclideanJordanAlgebraOperator( P, P, fda_elt.matrix().transpose() ) def quadratic_representation(self, other=None): """ Return the quadratic representation of this element. SETUP:: sage: from mjo.eja.eja_algebra import (JordanSpinEJA, ....: random_eja) EXAMPLES: The explicit form in the spin factor algebra is given by Alizadeh's Example 11.12:: sage: set_random_seed() sage: n = ZZ.random_element(1,10) sage: J = JordanSpinEJA(n) sage: x = J.random_element() sage: x_vec = x.vector() sage: x0 = x_vec[0] sage: x_bar = x_vec[1:] sage: A = matrix(QQ, 1, [x_vec.inner_product(x_vec)]) sage: B = 2*x0*x_bar.row() sage: C = 2*x0*x_bar.column() sage: D = matrix.identity(QQ, n-1) sage: D = (x0^2 - x_bar.inner_product(x_bar))*D sage: D = D + 2*x_bar.tensor_product(x_bar) sage: Q = matrix.block(2,2,[A,B,C,D]) sage: Q == x.quadratic_representation().matrix() True Test all of the properties from Theorem 11.2 in Alizadeh:: sage: set_random_seed() sage: J = random_eja() sage: x = J.random_element() sage: y = J.random_element() sage: Lx = x.operator() sage: Lxx = (x*x).operator() sage: Qx = x.quadratic_representation() sage: Qy = y.quadratic_representation() sage: Qxy = x.quadratic_representation(y) sage: Qex = J.one().quadratic_representation(x) sage: n = ZZ.random_element(10) sage: Qxn = (x^n).quadratic_representation() Property 1: sage: 2*Qxy == (x+y).quadratic_representation() - Qx - Qy True Property 2 (multiply on the right for :trac:`28272`): sage: alpha = QQ.random_element() sage: (alpha*x).quadratic_representation() == Qx*(alpha^2) True Property 3: sage: not x.is_invertible() or ( Qx(x.inverse()) == x ) True sage: not x.is_invertible() or ( ....: ~Qx ....: == ....: x.inverse().quadratic_representation() ) True sage: Qxy(J.one()) == x*y True Property 4: sage: not x.is_invertible() or ( ....: x.quadratic_representation(x.inverse())*Qx ....: == Qx*x.quadratic_representation(x.inverse()) ) True sage: not x.is_invertible() or ( ....: x.quadratic_representation(x.inverse())*Qx ....: == ....: 2*x.operator()*Qex - Qx ) True sage: 2*x.operator()*Qex - Qx == Lxx True Property 5: sage: Qy(x).quadratic_representation() == Qy*Qx*Qy True Property 6: sage: Qxn == (Qx)^n True Property 7: sage: not x.is_invertible() or ( ....: Qx*x.inverse().operator() == Lx ) True Property 8: sage: not x.operator_commutes_with(y) or ( ....: Qx(y)^n == Qxn(y^n) ) True """ if other is None: other=self elif not other in self.parent(): raise TypeError("'other' must live in the same algebra") L = self.operator() M = other.operator() return ( L*M + M*L - (self*other).operator() ) def span_of_powers(self): """ Return the vector space spanned by successive powers of this element. """ # The dimension of the subalgebra can't be greater than # the big algebra, so just put everything into a list # and let span() get rid of the excess. # # We do the extra ambient_vector_space() in case we're messing # with polynomials and the direct parent is a module. V = self.parent().vector_space() return V.span( (self**d).vector() for d in xrange(V.dimension()) ) def subalgebra_generated_by(self): """ Return the associative subalgebra of the parent EJA generated by this element. SETUP:: sage: from mjo.eja.eja_algebra import random_eja TESTS:: sage: set_random_seed() sage: x = random_eja().random_element() sage: x.subalgebra_generated_by().is_associative() True Squaring in the subalgebra should work the same as in the superalgebra:: sage: set_random_seed() sage: x = random_eja().random_element() sage: u = x.subalgebra_generated_by().random_element() sage: u.operator()(u) == u^2 True """ # First get the subspace spanned by the powers of myself... V = self.span_of_powers() F = self.base_ring() # Now figure out the entries of the right-multiplication # matrix for the successive basis elements b0, b1,... of # that subspace. mats = [] for b_right in V.basis(): eja_b_right = self.parent()(b_right) b_right_rows = [] # The first row of the right-multiplication matrix by # b1 is what we get if we apply that matrix to b1. The # second row of the right multiplication matrix by b1 # is what we get when we apply that matrix to b2... # # IMPORTANT: this assumes that all vectors are COLUMN # vectors, unlike our superclass (which uses row vectors). for b_left in V.basis(): eja_b_left = self.parent()(b_left) # Multiply in the original EJA, but then get the # coordinates from the subalgebra in terms of its # basis. this_row = V.coordinates((eja_b_left*eja_b_right).vector()) b_right_rows.append(this_row) b_right_matrix = matrix(F, b_right_rows) mats.append(b_right_matrix) # It's an algebra of polynomials in one element, and EJAs # are power-associative. # # TODO: choose generator names intelligently. return FiniteDimensionalEuclideanJordanAlgebra(F, mats, assume_associative=True, names='f') def subalgebra_idempotent(self): """ Find an idempotent in the associative subalgebra I generate using Proposition 2.3.5 in Baes. SETUP:: sage: from mjo.eja.eja_algebra import random_eja TESTS:: sage: set_random_seed() sage: J = random_eja() sage: x = J.random_element() sage: while x.is_nilpotent(): ....: x = J.random_element() sage: c = x.subalgebra_idempotent() sage: c^2 == c True """ if self.is_nilpotent(): raise ValueError("this only works with non-nilpotent elements!") V = self.span_of_powers() J = self.subalgebra_generated_by() # Mis-design warning: the basis used for span_of_powers() # and subalgebra_generated_by() must be the same, and in # the same order! u = J(V.coordinates(self.vector())) # The image of the matrix of left-u^m-multiplication # will be minimal for some natural number s... s = 0 minimal_dim = V.dimension() for i in xrange(1, V.dimension()): this_dim = (u**i).operator().matrix().image().dimension() if this_dim < minimal_dim: minimal_dim = this_dim s = i # Now minimal_matrix should correspond to the smallest # non-zero subspace in Baes's (or really, Koecher's) # proposition. # # However, we need to restrict the matrix to work on the # subspace... or do we? Can't we just solve, knowing that # A(c) = u^(s+1) should have a solution in the big space, # too? # # Beware, solve_right() means that we're using COLUMN vectors. # Our FiniteDimensionalAlgebraElement superclass uses rows. u_next = u**(s+1) A = u_next.operator().matrix() c_coordinates = A.solve_right(u_next.vector()) # Now c_coordinates is the idempotent we want, but it's in # the coordinate system of the subalgebra. # # We need the basis for J, but as elements of the parent algebra. # basis = [self.parent(v) for v in V.basis()] return self.parent().linear_combination(zip(c_coordinates, basis)) def trace(self): """ Return my trace, the sum of my eigenvalues. SETUP:: sage: from mjo.eja.eja_algebra import (JordanSpinEJA, ....: RealCartesianProductEJA, ....: random_eja) EXAMPLES:: sage: J = JordanSpinEJA(3) sage: x = sum(J.gens()) sage: x.trace() 2 :: sage: J = RealCartesianProductEJA(5) sage: J.one().trace() 5 TESTS: The trace of an element is a real number:: sage: set_random_seed() sage: J = random_eja() sage: J.random_element().trace() in J.base_ring() True """ P = self.parent() r = P.rank() p = P._charpoly_coeff(r-1) # The _charpoly_coeff function already adds the factor of # -1 to ensure that _charpoly_coeff(r-1) is really what # appears in front of t^{r-1} in the charpoly. However, # we want the negative of THAT for the trace. return -p(*self.vector()) def trace_inner_product(self, other): """ Return the trace inner product of myself and ``other``. SETUP:: sage: from mjo.eja.eja_algebra import random_eja TESTS: The trace inner product is commutative:: sage: set_random_seed() sage: J = random_eja() sage: x = J.random_element(); y = J.random_element() sage: x.trace_inner_product(y) == y.trace_inner_product(x) True The trace inner product is bilinear:: sage: set_random_seed() sage: J = random_eja() sage: x = J.random_element() sage: y = J.random_element() sage: z = J.random_element() sage: a = QQ.random_element(); sage: actual = (a*(x+z)).trace_inner_product(y) sage: expected = ( a*x.trace_inner_product(y) + ....: a*z.trace_inner_product(y) ) sage: actual == expected True sage: actual = x.trace_inner_product(a*(y+z)) sage: expected = ( a*x.trace_inner_product(y) + ....: a*x.trace_inner_product(z) ) sage: actual == expected True The trace inner product satisfies the compatibility condition in the definition of a Euclidean Jordan algebra:: sage: set_random_seed() sage: J = random_eja() sage: x = J.random_element() sage: y = J.random_element() sage: z = J.random_element() sage: (x*y).trace_inner_product(z) == y.trace_inner_product(x*z) True """ if not other in self.parent(): raise TypeError("'other' must live in the same algebra") return (self*other).trace() class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra): """ Return the Euclidean Jordan Algebra corresponding to the set `R^n` under the Hadamard product. Note: this is nothing more than the Cartesian product of ``n`` copies of the spin algebra. Once Cartesian product algebras are implemented, this can go. SETUP:: sage: from mjo.eja.eja_algebra import RealCartesianProductEJA EXAMPLES: This multiplication table can be verified by hand:: sage: J = RealCartesianProductEJA(3) sage: e0,e1,e2 = J.gens() sage: e0*e0 e0 sage: e0*e1 0 sage: e0*e2 0 sage: e1*e1 e1 sage: e1*e2 0 sage: e2*e2 e2 """ @staticmethod def __classcall_private__(cls, n, field=QQ): # The FiniteDimensionalAlgebra constructor takes a list of # matrices, the ith representing right multiplication by the ith # basis element in the vector space. So if e_1 = (1,0,0), then # right (Hadamard) multiplication of x by e_1 picks out the first # component of x; and likewise for the ith basis element e_i. Qs = [ matrix(field, n, n, lambda k,j: 1*(k == j == i)) for i in xrange(n) ] fdeja = super(RealCartesianProductEJA, cls) return fdeja.__classcall_private__(cls, field, Qs, rank=n) def inner_product(self, x, y): return _usual_ip(x,y) def random_eja(): """ Return a "random" finite-dimensional Euclidean Jordan Algebra. ALGORITHM: For now, we choose a random natural number ``n`` (greater than zero) and then give you back one of the following: * The cartesian product of the rational numbers ``n`` times; this is ``QQ^n`` with the Hadamard product. * The Jordan spin algebra on ``QQ^n``. * The ``n``-by-``n`` rational symmetric matrices with the symmetric product. * The ``n``-by-``n`` complex-rational Hermitian matrices embedded in the space of ``2n``-by-``2n`` real symmetric matrices. * The ``n``-by-``n`` quaternion-rational Hermitian matrices embedded in the space of ``4n``-by-``4n`` real symmetric matrices. Later this might be extended to return Cartesian products of the EJAs above. SETUP:: sage: from mjo.eja.eja_algebra import random_eja TESTS:: sage: random_eja() Euclidean Jordan algebra of degree... """ # The max_n component lets us choose different upper bounds on the # value "n" that gets passed to the constructor. This is needed # because e.g. R^{10} is reasonable to test, while the Hermitian # 10-by-10 quaternion matrices are not. (constructor, max_n) = choice([(RealCartesianProductEJA, 6), (JordanSpinEJA, 6), (RealSymmetricEJA, 5), (ComplexHermitianEJA, 4), (QuaternionHermitianEJA, 3)]) n = ZZ.random_element(1, max_n) return constructor(n, field=QQ) def _real_symmetric_basis(n, field=QQ): """ Return a basis for the space of real symmetric n-by-n matrices. """ # The basis of symmetric matrices, as matrices, in their R^(n-by-n) # coordinates. S = [] for i in xrange(n): for j in xrange(i+1): Eij = matrix(field, n, lambda k,l: k==i and l==j) if i == j: Sij = Eij else: # Beware, orthogonal but not normalized! Sij = Eij + Eij.transpose() S.append(Sij) return tuple(S) def _complex_hermitian_basis(n, field=QQ): """ Returns a basis for the space of complex Hermitian n-by-n matrices. SETUP:: sage: from mjo.eja.eja_algebra import _complex_hermitian_basis TESTS:: sage: set_random_seed() sage: n = ZZ.random_element(1,5) sage: all( M.is_symmetric() for M in _complex_hermitian_basis(n) ) True """ F = QuadraticField(-1, 'I') I = F.gen() # This is like the symmetric case, but we need to be careful: # # * We want conjugate-symmetry, not just symmetry. # * The diagonal will (as a result) be real. # S = [] for i in xrange(n): for j in xrange(i+1): Eij = matrix(field, n, lambda k,l: k==i and l==j) if i == j: Sij = _embed_complex_matrix(Eij) S.append(Sij) else: # Beware, orthogonal but not normalized! The second one # has a minus because it's conjugated. Sij_real = _embed_complex_matrix(Eij + Eij.transpose()) S.append(Sij_real) Sij_imag = _embed_complex_matrix(I*Eij - I*Eij.transpose()) S.append(Sij_imag) return tuple(S) def _quaternion_hermitian_basis(n, field=QQ): """ Returns a basis for the space of quaternion Hermitian n-by-n matrices. SETUP:: sage: from mjo.eja.eja_algebra import _quaternion_hermitian_basis TESTS:: sage: set_random_seed() sage: n = ZZ.random_element(1,5) sage: all( M.is_symmetric() for M in _quaternion_hermitian_basis(n) ) True """ Q = QuaternionAlgebra(QQ,-1,-1) I,J,K = Q.gens() # This is like the symmetric case, but we need to be careful: # # * We want conjugate-symmetry, not just symmetry. # * The diagonal will (as a result) be real. # S = [] for i in xrange(n): for j in xrange(i+1): Eij = matrix(Q, n, lambda k,l: k==i and l==j) if i == j: Sij = _embed_quaternion_matrix(Eij) S.append(Sij) else: # Beware, orthogonal but not normalized! The second, # third, and fourth ones have a minus because they're # conjugated. Sij_real = _embed_quaternion_matrix(Eij + Eij.transpose()) S.append(Sij_real) Sij_I = _embed_quaternion_matrix(I*Eij - I*Eij.transpose()) S.append(Sij_I) Sij_J = _embed_quaternion_matrix(J*Eij - J*Eij.transpose()) S.append(Sij_J) Sij_K = _embed_quaternion_matrix(K*Eij - K*Eij.transpose()) S.append(Sij_K) return tuple(S) def _mat2vec(m): return vector(m.base_ring(), m.list()) def _vec2mat(v): return matrix(v.base_ring(), sqrt(v.degree()), v.list()) def _multiplication_table_from_matrix_basis(basis): """ At least three of the five simple Euclidean Jordan algebras have the symmetric multiplication (A,B) |-> (AB + BA)/2, where the multiplication on the right is matrix multiplication. Given a basis for the underlying matrix space, this function returns a multiplication table (obtained by looping through the basis elements) for an algebra of those matrices. A reordered copy of the basis is also returned to work around the fact that the ``span()`` in this function will change the order of the basis from what we think it is, to... something else. """ # In S^2, for example, we nominally have four coordinates even # though the space is of dimension three only. The vector space V # is supposed to hold the entire long vector, and the subspace W # of V will be spanned by the vectors that arise from symmetric # matrices. Thus for S^2, dim(V) == 4 and dim(W) == 3. field = basis[0].base_ring() dimension = basis[0].nrows() V = VectorSpace(field, dimension**2) W = V.span( _mat2vec(s) for s in basis ) # Taking the span above reorders our basis (thanks, jerk!) so we # need to put our "matrix basis" in the same order as the # (reordered) vector basis. S = tuple( _vec2mat(b) for b in W.basis() ) Qs = [] for s in S: # Brute force the multiplication-by-s matrix by looping # through all elements of the basis and doing the computation # to find out what the corresponding row should be. BEWARE: # these multiplication tables won't be symmetric! It therefore # becomes REALLY IMPORTANT that the underlying algebra # constructor uses ROW vectors and not COLUMN vectors. That's # why we're computing rows here and not columns. Q_rows = [] for t in S: this_row = _mat2vec((s*t + t*s)/2) Q_rows.append(W.coordinates(this_row)) Q = matrix(field, W.dimension(), Q_rows) Qs.append(Q) return (Qs, S) def _embed_complex_matrix(M): """ Embed the n-by-n complex matrix ``M`` into the space of real matrices of size 2n-by-2n via the map the sends each entry `z = a + bi` to the block matrix ``[[a,b],[-b,a]]``. SETUP:: sage: from mjo.eja.eja_algebra import _embed_complex_matrix EXAMPLES:: sage: F = QuadraticField(-1,'i') sage: x1 = F(4 - 2*i) sage: x2 = F(1 + 2*i) sage: x3 = F(-i) sage: x4 = F(6) sage: M = matrix(F,2,[[x1,x2],[x3,x4]]) sage: _embed_complex_matrix(M) [ 4 -2| 1 2] [ 2 4|-2 1] [-----+-----] [ 0 -1| 6 0] [ 1 0| 0 6] TESTS: Embedding is a homomorphism (isomorphism, in fact):: sage: set_random_seed() sage: n = ZZ.random_element(5) sage: F = QuadraticField(-1, 'i') sage: X = random_matrix(F, n) sage: Y = random_matrix(F, n) sage: actual = _embed_complex_matrix(X) * _embed_complex_matrix(Y) sage: expected = _embed_complex_matrix(X*Y) sage: actual == expected True """ n = M.nrows() if M.ncols() != n: raise ValueError("the matrix 'M' must be square") field = M.base_ring() blocks = [] for z in M.list(): a = z.real() b = z.imag() blocks.append(matrix(field, 2, [[a,b],[-b,a]])) # We can drop the imaginaries here. return matrix.block(field.base_ring(), n, blocks) def _unembed_complex_matrix(M): """ The inverse of _embed_complex_matrix(). SETUP:: sage: from mjo.eja.eja_algebra import (_embed_complex_matrix, ....: _unembed_complex_matrix) EXAMPLES:: sage: A = matrix(QQ,[ [ 1, 2, 3, 4], ....: [-2, 1, -4, 3], ....: [ 9, 10, 11, 12], ....: [-10, 9, -12, 11] ]) sage: _unembed_complex_matrix(A) [ 2*i + 1 4*i + 3] [ 10*i + 9 12*i + 11] TESTS: Unembedding is the inverse of embedding:: sage: set_random_seed() sage: F = QuadraticField(-1, 'i') sage: M = random_matrix(F, 3) sage: _unembed_complex_matrix(_embed_complex_matrix(M)) == M True """ n = ZZ(M.nrows()) if M.ncols() != n: raise ValueError("the matrix 'M' must be square") if not n.mod(2).is_zero(): raise ValueError("the matrix 'M' must be a complex embedding") F = QuadraticField(-1, 'i') i = F.gen() # Go top-left to bottom-right (reading order), converting every # 2-by-2 block we see to a single complex element. elements = [] for k in xrange(n/2): for j in xrange(n/2): submat = M[2*k:2*k+2,2*j:2*j+2] if submat[0,0] != submat[1,1]: raise ValueError('bad on-diagonal submatrix') if submat[0,1] != -submat[1,0]: raise ValueError('bad off-diagonal submatrix') z = submat[0,0] + submat[0,1]*i elements.append(z) return matrix(F, n/2, elements) def _embed_quaternion_matrix(M): """ Embed the n-by-n quaternion matrix ``M`` into the space of real matrices of size 4n-by-4n by first sending each quaternion entry `z = a + bi + cj + dk` to the block-complex matrix ``[[a + bi, c+di],[-c + di, a-bi]]`, and then embedding those into a real matrix. SETUP:: sage: from mjo.eja.eja_algebra import _embed_quaternion_matrix EXAMPLES:: sage: Q = QuaternionAlgebra(QQ,-1,-1) sage: i,j,k = Q.gens() sage: x = 1 + 2*i + 3*j + 4*k sage: M = matrix(Q, 1, [[x]]) sage: _embed_quaternion_matrix(M) [ 1 2 3 4] [-2 1 -4 3] [-3 4 1 -2] [-4 -3 2 1] Embedding is a homomorphism (isomorphism, in fact):: sage: set_random_seed() sage: n = ZZ.random_element(5) sage: Q = QuaternionAlgebra(QQ,-1,-1) sage: X = random_matrix(Q, n) sage: Y = random_matrix(Q, n) sage: actual = _embed_quaternion_matrix(X)*_embed_quaternion_matrix(Y) sage: expected = _embed_quaternion_matrix(X*Y) sage: actual == expected True """ quaternions = M.base_ring() n = M.nrows() if M.ncols() != n: raise ValueError("the matrix 'M' must be square") F = QuadraticField(-1, 'i') i = F.gen() blocks = [] for z in M.list(): t = z.coefficient_tuple() a = t[0] b = t[1] c = t[2] d = t[3] cplx_matrix = matrix(F, 2, [[ a + b*i, c + d*i], [-c + d*i, a - b*i]]) blocks.append(_embed_complex_matrix(cplx_matrix)) # We should have real entries by now, so use the realest field # we've got for the return value. return matrix.block(quaternions.base_ring(), n, blocks) def _unembed_quaternion_matrix(M): """ The inverse of _embed_quaternion_matrix(). SETUP:: sage: from mjo.eja.eja_algebra import (_embed_quaternion_matrix, ....: _unembed_quaternion_matrix) EXAMPLES:: sage: M = matrix(QQ, [[ 1, 2, 3, 4], ....: [-2, 1, -4, 3], ....: [-3, 4, 1, -2], ....: [-4, -3, 2, 1]]) sage: _unembed_quaternion_matrix(M) [1 + 2*i + 3*j + 4*k] TESTS: Unembedding is the inverse of embedding:: sage: set_random_seed() sage: Q = QuaternionAlgebra(QQ, -1, -1) sage: M = random_matrix(Q, 3) sage: _unembed_quaternion_matrix(_embed_quaternion_matrix(M)) == M True """ n = ZZ(M.nrows()) if M.ncols() != n: raise ValueError("the matrix 'M' must be square") if not n.mod(4).is_zero(): raise ValueError("the matrix 'M' must be a complex embedding") Q = QuaternionAlgebra(QQ,-1,-1) i,j,k = Q.gens() # Go top-left to bottom-right (reading order), converting every # 4-by-4 block we see to a 2-by-2 complex block, to a 1-by-1 # quaternion block. elements = [] for l in xrange(n/4): for m in xrange(n/4): submat = _unembed_complex_matrix(M[4*l:4*l+4,4*m:4*m+4]) if submat[0,0] != submat[1,1].conjugate(): raise ValueError('bad on-diagonal submatrix') if submat[0,1] != -submat[1,0].conjugate(): raise ValueError('bad off-diagonal submatrix') z = submat[0,0].real() + submat[0,0].imag()*i z += submat[0,1].real()*j + submat[0,1].imag()*k elements.append(z) return matrix(Q, n/4, elements) # The usual inner product on R^n. def _usual_ip(x,y): return x.vector().inner_product(y.vector()) # The inner product used for the real symmetric simple EJA. # We keep it as a separate function because e.g. the complex # algebra uses the same inner product, except divided by 2. def _matrix_ip(X,Y): X_mat = X.natural_representation() Y_mat = Y.natural_representation() return (X_mat*Y_mat).trace() class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra): """ The rank-n simple EJA consisting of real symmetric n-by-n matrices, the usual symmetric Jordan product, and the trace inner product. It has dimension `(n^2 + n)/2` over the reals. SETUP:: sage: from mjo.eja.eja_algebra import RealSymmetricEJA EXAMPLES:: sage: J = RealSymmetricEJA(2) sage: e0, e1, e2 = J.gens() sage: e0*e0 e0 sage: e1*e1 e0 + e2 sage: e2*e2 e2 TESTS: The degree of this algebra is `(n^2 + n) / 2`:: sage: set_random_seed() sage: n = ZZ.random_element(1,5) sage: J = RealSymmetricEJA(n) sage: J.degree() == (n^2 + n)/2 True The Jordan multiplication is what we think it is:: sage: set_random_seed() sage: n = ZZ.random_element(1,5) sage: J = RealSymmetricEJA(n) sage: x = J.random_element() sage: y = J.random_element() sage: actual = (x*y).natural_representation() sage: X = x.natural_representation() sage: Y = y.natural_representation() sage: expected = (X*Y + Y*X)/2 sage: actual == expected True sage: J(expected) == x*y True """ @staticmethod def __classcall_private__(cls, n, field=QQ): S = _real_symmetric_basis(n, field=field) (Qs, T) = _multiplication_table_from_matrix_basis(S) fdeja = super(RealSymmetricEJA, cls) return fdeja.__classcall_private__(cls, field, Qs, rank=n, natural_basis=T) def inner_product(self, x, y): return _matrix_ip(x,y) class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra): """ 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 TESTS: The degree of this algebra is `n^2`:: sage: set_random_seed() sage: n = ZZ.random_element(1,5) sage: J = ComplexHermitianEJA(n) sage: J.degree() == n^2 True The Jordan multiplication is what we think it is:: sage: set_random_seed() sage: n = ZZ.random_element(1,5) sage: J = ComplexHermitianEJA(n) sage: x = J.random_element() sage: y = J.random_element() sage: actual = (x*y).natural_representation() sage: X = x.natural_representation() sage: Y = y.natural_representation() sage: expected = (X*Y + Y*X)/2 sage: actual == expected True sage: J(expected) == x*y True """ @staticmethod def __classcall_private__(cls, n, field=QQ): S = _complex_hermitian_basis(n) (Qs, T) = _multiplication_table_from_matrix_basis(S) fdeja = super(ComplexHermitianEJA, cls) return fdeja.__classcall_private__(cls, field, Qs, rank=n, natural_basis=T) def inner_product(self, x, y): # Since a+bi on the diagonal is represented as # # a + bi = [ a b ] # [ -b a ], # # we'll double-count the "a" entries if we take the trace of # the embedding. return _matrix_ip(x,y)/2 class QuaternionHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra): """ 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 TESTS: The degree of this algebra is `n^2`:: sage: set_random_seed() sage: n = ZZ.random_element(1,5) sage: J = QuaternionHermitianEJA(n) sage: J.degree() == 2*(n^2) - n True The Jordan multiplication is what we think it is:: sage: set_random_seed() sage: n = ZZ.random_element(1,5) sage: J = QuaternionHermitianEJA(n) sage: x = J.random_element() sage: y = J.random_element() sage: actual = (x*y).natural_representation() sage: X = x.natural_representation() sage: Y = y.natural_representation() sage: expected = (X*Y + Y*X)/2 sage: actual == expected True sage: J(expected) == x*y True """ @staticmethod def __classcall_private__(cls, n, field=QQ): S = _quaternion_hermitian_basis(n) (Qs, T) = _multiplication_table_from_matrix_basis(S) fdeja = super(QuaternionHermitianEJA, cls) return fdeja.__classcall_private__(cls, field, Qs, rank=n, natural_basis=T) def inner_product(self, x, y): # Since a+bi+cj+dk on the diagonal is represented as # # a + bi +cj + dk = [ a b c d] # [ -b a -d c] # [ -c d a -b] # [ -d -c b a], # # we'll quadruple-count the "a" entries if we take the trace of # the embedding. return _matrix_ip(x,y)/4 class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra): """ The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)`` with the usual inner product and jordan product ``x*y = (, x0*y_bar + y0*x_bar)``. It has dimension `n` over the reals. SETUP:: sage: from mjo.eja.eja_algebra import JordanSpinEJA EXAMPLES: This multiplication table can be verified by hand:: sage: J = JordanSpinEJA(4) sage: e0,e1,e2,e3 = J.gens() sage: e0*e0 e0 sage: e0*e1 e1 sage: e0*e2 e2 sage: e0*e3 e3 sage: e1*e2 0 sage: e1*e3 0 sage: e2*e3 0 """ @staticmethod def __classcall_private__(cls, n, field=QQ): Qs = [] id_matrix = matrix.identity(field, n) for i in xrange(n): ei = id_matrix.column(i) Qi = matrix.zero(field, n) Qi.set_row(0, ei) Qi.set_column(0, ei) Qi += matrix.diagonal(n, [ei[0]]*n) # The addition of the diagonal matrix adds an extra ei[0] in the # upper-left corner of the matrix. Qi[0,0] = Qi[0,0] * ~field(2) Qs.append(Qi) # The rank of the spin algebra is two, unless we're in a # one-dimensional ambient space (because the rank is bounded by # the ambient dimension). fdeja = super(JordanSpinEJA, cls) return fdeja.__classcall_private__(cls, field, Qs, rank=min(n,2)) def inner_product(self, x, y): return _usual_ip(x,y)