X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feuclidean_jordan_algebra.py;h=d6235d33ee2edd77f24d1b304e8d4d80b4342d9f;hb=5f2deba7b079cfe5a0a290f810e569bfb480d186;hp=fdaccba58a8b99a2f5222054358969ce3e731882;hpb=cef21b24d30d942dbaa542a23aab642c884371f7;p=sage.d.git diff --git a/mjo/eja/euclidean_jordan_algebra.py b/mjo/eja/euclidean_jordan_algebra.py index fdaccba..d6235d3 100644 --- a/mjo/eja/euclidean_jordan_algebra.py +++ b/mjo/eja/euclidean_jordan_algebra.py @@ -133,12 +133,12 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): EXAMPLES:: - sage: J = eja_ln(2) + sage: J = JordanSpinSimpleEJA(2) sage: e0,e1 = J.gens() sage: x = e0 + e1 sage: x.det() 0 - sage: J = eja_ln(3) + sage: J = JordanSpinSimpleEJA(3) sage: e0,e1,e2 = J.gens() sage: x = e0 + e1 + e2 sage: x.det() @@ -207,7 +207,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): The identity element always has degree one, but any element linearly-independent from it is regular:: - sage: J = eja_ln(5) + sage: J = JordanSpinSimpleEJA(5) sage: J.one().is_regular() False sage: e0, e1, e2, e3, e4 = J.gens() # e0 is the identity @@ -232,7 +232,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): EXAMPLES:: - sage: J = eja_ln(4) + sage: J = JordanSpinSimpleEJA(4) sage: J.one().degree() 1 sage: e0,e1,e2,e3 = J.gens() @@ -244,7 +244,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): sage: set_random_seed() sage: n = ZZ.random_element(1,10).abs() - sage: J = eja_ln(n) + sage: J = JordanSpinSimpleEJA(n) sage: x = J.random_element() sage: x == x.coefficient(0)*J.one() or x.degree() == 2 True @@ -289,7 +289,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): sage: set_random_seed() sage: n = ZZ.random_element(2,10).abs() - sage: J = eja_ln(n) + sage: J = JordanSpinSimpleEJA(n) sage: y = J.random_element() sage: while y == y.coefficient(0)*J.one(): ....: y = J.random_element() @@ -333,7 +333,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): Alizadeh's Example 11.12:: sage: n = ZZ.random_element(1,10).abs() - sage: J = eja_ln(n) + sage: J = JordanSpinSimpleEJA(n) sage: x = J.random_element() sage: x_vec = x.vector() sage: x0 = x_vec[0] @@ -433,7 +433,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): sage: c = J.random_element().subalgebra_idempotent() sage: c^2 == c True - sage: J = eja_ln(5) + sage: J = JordanSpinSimpleEJA(5) sage: c = J.random_element().subalgebra_idempotent() sage: c^2 == c True @@ -489,7 +489,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): EXAMPLES:: - sage: J = eja_ln(3) + sage: J = JordanSpinSimpleEJA(3) sage: e0,e1,e2 = J.gens() sage: x = e0 + e1 + e2 sage: x.trace() @@ -549,82 +549,6 @@ def eja_rn(dimension, field=QQ): return FiniteDimensionalEuclideanJordanAlgebra(field,Qs,rank=dimension) -def eja_ln(dimension, field=QQ): - """ - Return the Jordan algebra corresponding to the Lorentz "ice cream" - cone of the given ``dimension``. - - EXAMPLES: - - This multiplication table can be verified by hand:: - - sage: J = eja_ln(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 - - In one dimension, this is the reals under multiplication:: - - sage: J1 = eja_ln(1) - sage: J2 = eja_rn(1) - sage: J1 == J2 - True - - """ - Qs = [] - id_matrix = identity_matrix(field,dimension) - for i in xrange(dimension): - ei = id_matrix.column(i) - Qi = zero_matrix(field,dimension) - Qi.set_row(0, ei) - Qi.set_column(0, ei) - Qi += diagonal_matrix(dimension, [ei[0]]*dimension) - # 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). - rank = min(dimension,2) - return FiniteDimensionalEuclideanJordanAlgebra(field,Qs,rank=rank) - - -def eja_sn(dimension, field=QQ): - """ - Return the simple Jordan algebra of ``dimension``-by-``dimension`` - symmetric matrices over ``field``. - - EXAMPLES:: - - sage: J = eja_sn(2) - sage: e0, e1, e2 = J.gens() - sage: e0*e0 - e0 - sage: e1*e1 - e0 + e2 - sage: e2*e2 - e2 - - """ - S = _real_symmetric_basis(dimension, field=field) - Qs = _multiplication_table_from_matrix_basis(S) - - return FiniteDimensionalEuclideanJordanAlgebra(field,Qs,rank=dimension) - def random_eja(): """ @@ -652,9 +576,12 @@ def random_eja(): Euclidean Jordan algebra of degree... """ - n = ZZ.random_element(1,10).abs() - constructor = choice([eja_rn, eja_ln, eja_sn]) - return constructor(dimension=n, field=QQ) + n = ZZ.random_element(1,5).abs() + constructor = choice([eja_rn, + JordanSpinSimpleEJA, + RealSymmetricSimpleEJA, + ComplexHermitianSimpleEJA]) + return constructor(n, field=QQ) @@ -677,6 +604,43 @@ def _real_symmetric_basis(n, field=QQ): return S +def _complex_hermitian_basis(n, field=QQ): + """ + Returns a basis for the space of complex Hermitian n-by-n matrices. + + TESTS:: + + sage: set_random_seed() + sage: n = ZZ.random_element(1,5).abs() + 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 S + + def _multiplication_table_from_matrix_basis(basis): """ At least three of the five simple Euclidean Jordan algebras have the @@ -709,7 +673,7 @@ def _multiplication_table_from_matrix_basis(basis): S = [ vec2mat(b) for b in W.basis() ] Qs = [] - for s in basis: + 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: @@ -718,10 +682,10 @@ def _multiplication_table_from_matrix_basis(basis): # constructor uses ROW vectors and not COLUMN vectors. That's # why we're computing rows here and not columns. Q_rows = [] - for t in basis: + for t in S: this_row = mat2vec((s*t + t*s)/2) Q_rows.append(W.coordinates(this_row)) - Q = matrix(field,Q_rows) + Q = matrix(field, W.dimension(), Q_rows) Qs.append(Q) return Qs @@ -758,25 +722,106 @@ def _embed_complex_matrix(M): a = z.real() b = z.imag() blocks.append(matrix(field, 2, [[a,-b],[b,a]])) - return block_matrix(field, n, blocks) + # We can drop the imaginaries here. + return block_matrix(field.base_ring(), n, blocks) + + +def _unembed_complex_matrix(M): + """ + The inverse of _embed_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] + """ + n = ZZ(M.nrows()) + if M.ncols() != n: + raise ArgumentError("the matrix 'M' must be square") + if not n.mod(2).is_zero(): + raise ArgumentError("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 ArgumentError('bad real submatrix') + if submat[0,1] != -submat[1,0]: + raise ArgumentError('bad imag submatrix') + z = submat[0,0] + submat[1,0]*i + elements.append(z) + + return matrix(F, n/2, elements) -def RealSymmetricSimpleEJA(n): + +def RealSymmetricSimpleEJA(n, field=QQ): """ 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. + + EXAMPLES:: + + sage: J = RealSymmetricSimpleEJA(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).abs() + sage: J = RealSymmetricSimpleEJA(n) + sage: J.degree() == (n^2 + n)/2 + True + """ - pass + S = _real_symmetric_basis(n, field=field) + Qs = _multiplication_table_from_matrix_basis(S) + + return FiniteDimensionalEuclideanJordanAlgebra(field,Qs,rank=n) -def ComplexHermitianSimpleEJA(n): + +def ComplexHermitianSimpleEJA(n, field=QQ): """ 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 + and the real-part-of-trace inner product. It has dimension `n^2` over the reals. + + TESTS: + + The degree of this algebra is `n^2`:: + + sage: set_random_seed() + sage: n = ZZ.random_element(1,5).abs() + sage: J = ComplexHermitianSimpleEJA(n) + sage: J.degree() == n^2 + True + """ - pass + S = _complex_hermitian_basis(n) + Qs = _multiplication_table_from_matrix_basis(S) + return FiniteDimensionalEuclideanJordanAlgebra(field, Qs, rank=n) + def QuaternionHermitianSimpleEJA(n): """ @@ -794,11 +839,56 @@ def OctonionHermitianSimpleEJA(n): n = 3 pass -def JordanSpinSimpleEJA(n): +def JordanSpinSimpleEJA(n, field=QQ): """ 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. + + EXAMPLES: + + This multiplication table can be verified by hand:: + + sage: J = JordanSpinSimpleEJA(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 + + In one dimension, this is the reals under multiplication:: + + sage: J1 = JordanSpinSimpleEJA(1) + sage: J2 = eja_rn(1) + sage: J1 == J2 + True + """ - pass + 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))