X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feuclidean_jordan_algebra.py;h=15ff26ca909cd1fbec4a2f1fc95e69b69f5bd02f;hb=b82bd087507b8f727c10ca0eb55f9a1d15ed3438;hp=1e5ada2188c3b6b3834165f221ab20b56d5c1f98;hpb=f2564c0412a2498ff32245a848ba61746124eaf8;p=sage.d.git diff --git a/mjo/eja/euclidean_jordan_algebra.py b/mjo/eja/euclidean_jordan_algebra.py index 1e5ada2..15ff26c 100644 --- a/mjo/eja/euclidean_jordan_algebra.py +++ b/mjo/eja/euclidean_jordan_algebra.py @@ -21,8 +21,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): assume_associative=False, category=None, rank=None, - natural_basis=None, - inner_product=None): + natural_basis=None): n = len(mult_table) mult_table = [b.base_extend(field) for b in mult_table] for b in mult_table: @@ -46,18 +45,17 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): names=names, category=cat, rank=rank, - natural_basis=natural_basis, - inner_product=inner_product) + natural_basis=natural_basis) - def __init__(self, field, + def __init__(self, + field, mult_table, names='e', assume_associative=False, category=None, rank=None, - natural_basis=None, - inner_product=None): + natural_basis=None): """ EXAMPLES: @@ -73,7 +71,6 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): """ self._rank = rank self._natural_basis = natural_basis - self._inner_product = inner_product fda = super(FiniteDimensionalEuclideanJordanAlgebra, self) fda.__init__(field, mult_table, @@ -93,7 +90,9 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): """ The inner product associated with this Euclidean Jordan algebra. - Will default to the trace inner product if nothing else. + Defaults to the trace inner product, but can be overridden by + subclasses if they are sure that the necessary properties are + satisfied. EXAMPLES: @@ -111,10 +110,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): """ if (not x in self) or (not y in self): raise TypeError("arguments must live in this algebra") - if self._inner_product is None: - return x.trace_inner_product(y) - else: - return self._inner_product(x,y) + return x.trace_inner_product(y) def natural_basis(self): @@ -132,7 +128,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): EXAMPLES:: - sage: J = RealSymmetricSimpleEJA(2) + sage: J = RealSymmetricEJA(2) sage: J.basis() Family (e0, e1, e2) sage: J.natural_basis() @@ -143,7 +139,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): :: - sage: J = JordanSpinSimpleEJA(2) + sage: J = JordanSpinEJA(2) sage: J.basis() Family (e0, e1) sage: J.natural_basis() @@ -180,14 +176,14 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): The identity in `S^n` is converted to the identity in the EJA:: - sage: J = RealSymmetricSimpleEJA(3) + sage: J = RealSymmetricEJA(3) sage: I = identity_matrix(QQ,3) sage: J(I) == J.one() True This skew-symmetric matrix can't be represented in the EJA:: - sage: J = RealSymmetricSimpleEJA(3) + sage: J = RealSymmetricEJA(3) sage: A = matrix(QQ,3, lambda i,j: i-j) sage: J(A) Traceback (most recent call last): @@ -295,7 +291,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): 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 = JordanSpinSimpleEJA(3) + sage: J = JordanSpinEJA(3) sage: x = vector(QQ,[1,2,3]) sage: y = vector(QQ,[4,5,6]) sage: x.inner_product(y) @@ -308,7 +304,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): so the inner product of the identity matrix with itself should be the `n`:: - sage: J = RealSymmetricSimpleEJA(3) + sage: J = RealSymmetricEJA(3) sage: J.one().inner_product(J.one()) 3 @@ -317,7 +313,13 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): part because the product of Hermitian matrices may not be Hermitian:: - sage: J = ComplexHermitianSimpleEJA(3) + 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 @@ -384,12 +386,12 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): EXAMPLES:: - sage: J = JordanSpinSimpleEJA(2) + sage: J = JordanSpinEJA(2) sage: e0,e1 = J.gens() sage: x = e0 + e1 sage: x.det() 0 - sage: J = JordanSpinSimpleEJA(3) + sage: J = JordanSpinEJA(3) sage: e0,e1,e2 = J.gens() sage: x = e0 + e1 + e2 sage: x.det() @@ -418,7 +420,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): sage: set_random_seed() sage: n = ZZ.random_element(1,10) - sage: J = JordanSpinSimpleEJA(n) + sage: J = JordanSpinEJA(n) sage: x = J.random_element() sage: while x.is_zero(): ....: x = J.random_element() @@ -548,7 +550,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): The identity element always has degree one, but any element linearly-independent from it is regular:: - sage: J = JordanSpinSimpleEJA(5) + sage: J = JordanSpinEJA(5) sage: J.one().is_regular() False sage: e0, e1, e2, e3, e4 = J.gens() # e0 is the identity @@ -573,7 +575,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): EXAMPLES:: - sage: J = JordanSpinSimpleEJA(4) + sage: J = JordanSpinEJA(4) sage: J.one().degree() 1 sage: e0,e1,e2,e3 = J.gens() @@ -585,7 +587,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): sage: set_random_seed() sage: n = ZZ.random_element(1,10) - sage: J = JordanSpinSimpleEJA(n) + sage: J = JordanSpinEJA(n) sage: x = J.random_element() sage: x == x.coefficient(0)*J.one() or x.degree() == 2 True @@ -617,7 +619,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): sage: set_random_seed() sage: n = ZZ.random_element(2,10) - sage: J = JordanSpinSimpleEJA(n) + sage: J = JordanSpinEJA(n) sage: y = J.random_element() sage: while y == y.coefficient(0)*J.one(): ....: y = J.random_element() @@ -663,7 +665,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): EXAMPLES:: - sage: J = ComplexHermitianSimpleEJA(3) + sage: J = ComplexHermitianEJA(3) sage: J.one() e0 + e5 + e8 sage: J.one().natural_representation() @@ -674,6 +676,25 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): [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() @@ -758,7 +779,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): sage: set_random_seed() sage: n = ZZ.random_element(1,10) - sage: J = JordanSpinSimpleEJA(n) + sage: J = JordanSpinEJA(n) sage: x = J.random_element() sage: x_vec = x.vector() sage: x0 = x_vec[0] @@ -901,11 +922,11 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): TESTS:: sage: set_random_seed() - sage: J = eja_rn(5) + sage: J = RealCartesianProductEJA(5) sage: c = J.random_element().subalgebra_idempotent() sage: c^2 == c True - sage: J = JordanSpinSimpleEJA(5) + sage: J = JordanSpinEJA(5) sage: c = J.random_element().subalgebra_idempotent() sage: c^2 == c True @@ -961,7 +982,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): EXAMPLES:: - sage: J = JordanSpinSimpleEJA(3) + sage: J = JordanSpinEJA(3) sage: e0,e1,e2 = J.gens() sage: x = e0 + e1 + e2 sage: x.trace() @@ -985,16 +1006,20 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): return (self*other).trace() -def eja_rn(dimension, field=QQ): +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. + EXAMPLES: This multiplication table can be verified by hand:: - sage: J = eja_rn(3) + sage: J = RealCartesianProductEJA(3) sage: e0,e1,e2 = J.gens() sage: e0*e0 e0 @@ -1010,19 +1035,21 @@ def eja_rn(dimension, field=QQ): e2 """ - # 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, dimension, dimension, lambda k,j: 1*(k == j == i)) - for i in xrange(dimension) ] - - return FiniteDimensionalEuclideanJordanAlgebra(field, - Qs, - rank=dimension, - inner_product=_usual_ip) + @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(): @@ -1042,6 +1069,12 @@ def random_eja(): * 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. @@ -1052,10 +1085,11 @@ def random_eja(): """ n = ZZ.random_element(1,5) - constructor = choice([eja_rn, - JordanSpinSimpleEJA, - RealSymmetricSimpleEJA, - ComplexHermitianSimpleEJA]) + constructor = choice([RealCartesianProductEJA, + JordanSpinEJA, + RealSymmetricEJA, + ComplexHermitianEJA, + QuaternionHermitianEJA]) return constructor(n, field=QQ) @@ -1116,6 +1150,48 @@ def _complex_hermitian_basis(n, field=QQ): return tuple(S) +def _quaternion_hermitian_basis(n, field=QQ): + """ + Returns a basis for the space of quaternion Hermitian n-by-n matrices. + + 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()) @@ -1190,6 +1266,20 @@ def _embed_complex_matrix(M): [ 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: @@ -1219,7 +1309,9 @@ def _unembed_complex_matrix(M): [ 2*i + 1 4*i + 3] [ 10*i + 9 12*i + 11] - TESTS:: + TESTS: + + Unembedding is the inverse of embedding:: sage: set_random_seed() sage: F = QuadraticField(-1, 'i') @@ -1244,14 +1336,123 @@ def _unembed_complex_matrix(M): 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 real submatrix') + raise ValueError('bad on-diagonal submatrix') if submat[0,1] != -submat[1,0]: - raise ValueError('bad imag submatrix') + 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. + + 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 block_matrix(quaternions.base_ring(), n, blocks) + + +def _unembed_quaternion_matrix(M): + """ + The inverse of _embed_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()) @@ -1265,7 +1466,7 @@ def _matrix_ip(X,Y): return (X_mat*Y_mat).trace() -def RealSymmetricSimpleEJA(n, field=QQ): +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 @@ -1273,7 +1474,7 @@ def RealSymmetricSimpleEJA(n, field=QQ): EXAMPLES:: - sage: J = RealSymmetricSimpleEJA(2) + sage: J = RealSymmetricEJA(2) sage: e0, e1, e2 = J.gens() sage: e0*e0 e0 @@ -1288,22 +1489,44 @@ def RealSymmetricSimpleEJA(n, field=QQ): sage: set_random_seed() sage: n = ZZ.random_element(1,5) - sage: J = RealSymmetricSimpleEJA(n) + 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 + """ - S = _real_symmetric_basis(n, field=field) - (Qs, T) = _multiplication_table_from_matrix_basis(S) + @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) - return FiniteDimensionalEuclideanJordanAlgebra(field, - Qs, - rank=n, - natural_basis=T, - inner_product=_matrix_ip) + def inner_product(self, x, y): + return _matrix_ip(x,y) -def ComplexHermitianSimpleEJA(n, field=QQ): +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, @@ -1316,47 +1539,110 @@ def ComplexHermitianSimpleEJA(n, field=QQ): sage: set_random_seed() sage: n = ZZ.random_element(1,5) - sage: J = ComplexHermitianSimpleEJA(n) + 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 + """ - S = _complex_hermitian_basis(n) - (Qs, T) = _multiplication_table_from_matrix_basis(S) + @staticmethod + def __classcall_private__(cls, n, field=QQ): + S = _complex_hermitian_basis(n) + (Qs, T) = _multiplication_table_from_matrix_basis(S) - # 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. - ip = lambda X,Y: _matrix_ip(X,Y)/2 + fdeja = super(ComplexHermitianEJA, cls) + return fdeja.__classcall_private__(cls, + field, + Qs, + rank=n, + natural_basis=T) - return FiniteDimensionalEuclideanJordanAlgebra(field, - Qs, - rank=n, - natural_basis=T, - inner_product=ip) + 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 -def QuaternionHermitianSimpleEJA(n): +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. - """ - pass -def OctonionHermitianSimpleEJA(n): - """ - This shit be crazy. It has dimension 27 over the reals. + 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 + """ - n = 3 - pass + @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 JordanSpinSimpleEJA(n, field=QQ): + 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 = @@ -1367,7 +1653,7 @@ def JordanSpinSimpleEJA(n, field=QQ): This multiplication table can be verified by hand:: - sage: J = JordanSpinSimpleEJA(4) + sage: J = JordanSpinEJA(4) sage: e0,e1,e2,e3 = J.gens() sage: e0*e0 e0 @@ -1384,31 +1670,27 @@ def JordanSpinSimpleEJA(n, field=QQ): sage: e2*e3 0 - In one dimension, this is the reals under multiplication:: - - sage: J1 = JordanSpinSimpleEJA(1) - sage: J2 = eja_rn(1) - sage: J1 == J2 - True - """ - Qs = [] - id_matrix = identity_matrix(field, n) - for i in xrange(n): - ei = id_matrix.column(i) - Qi = zero_matrix(field, n) - Qi.set_row(0, ei) - Qi.set_column(0, ei) - Qi += diagonal_matrix(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 factor algebra is two, UNLESS we're in a - # one-dimensional ambient space (the rank is bounded by the - # ambient dimension). - return FiniteDimensionalEuclideanJordanAlgebra(field, - Qs, - rank=min(n,2), - inner_product=_usual_ip) + @staticmethod + def __classcall_private__(cls, n, field=QQ): + Qs = [] + id_matrix = identity_matrix(field, n) + for i in xrange(n): + ei = id_matrix.column(i) + Qi = zero_matrix(field, n) + Qi.set_row(0, ei) + Qi.set_column(0, ei) + Qi += diagonal_matrix(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)