X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feuclidean_jordan_algebra.py;h=6bc53a172f5bd72ccc3cec7315bb908425160f57;hb=253d2f0d413e4ea64295242411f24673ae8e0b17;hp=d07425f6516e0261030afc3052ddbfd8660e6722;hpb=206683c90a5fd290ff4fd16a9ed272d9f981b668;p=sage.d.git diff --git a/mjo/eja/euclidean_jordan_algebra.py b/mjo/eja/euclidean_jordan_algebra.py index d07425f..6bc53a1 100644 --- a/mjo/eja/euclidean_jordan_algebra.py +++ b/mjo/eja/euclidean_jordan_algebra.py @@ -503,6 +503,16 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): raise ValueError('charpoly had fewer than 2 coefficients') + def trace_inner_product(self, other): + """ + Return the trace inner product of myself and ``other``. + """ + if not other in self.parent(): + raise ArgumentError("'other' must live in the same algebra") + + return (self*other).trace() + + def eja_rn(dimension, field=QQ): """ Return the Euclidean Jordan Algebra corresponding to the set @@ -610,27 +620,79 @@ def eja_sn(dimension, field=QQ): e2 """ - Qs = [] + S = _real_symmetric_basis(dimension, field=field) + Qs = _multiplication_table_from_matrix_basis(S) - # 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. - V = VectorSpace(field, dimension**2) + return FiniteDimensionalEuclideanJordanAlgebra(field,Qs,rank=dimension) + + +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. + + Later this might be extended to return Cartesian products of the + EJAs above. + + TESTS:: + + sage: random_eja() + Euclidean Jordan algebra of degree... + + """ + n = ZZ.random_element(1,5).abs() + constructor = choice([eja_rn, eja_ln, eja_sn, ComplexHermitianSimpleEJA]) + 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(dimension): + for i in xrange(n): for j in xrange(i+1): - Eij = matrix(field, dimension, lambda k,l: k==i and l==j) + 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 S + + +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. + """ + # 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() def mat2vec(m): return vector(field, m.list()) @@ -638,13 +700,15 @@ def eja_sn(dimension, field=QQ): def vec2mat(v): return matrix(field, dimension, v.list()) - W = V.span( mat2vec(s) for s in S ) + 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 = [ 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 @@ -657,38 +721,134 @@ def eja_sn(dimension, field=QQ): 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 FiniteDimensionalEuclideanJordanAlgebra(field,Qs,rank=dimension) + return Qs -def random_eja(): +def _embed_complex_matrix(M): """ - Return a "random" finite-dimensional Euclidean Jordan Algebra. + 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]]``. - ALGORITHM: + EXAMPLES:: - For now, we choose a random natural number ``n`` (greater than zero) - and then give you back one of the following: + 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] - * The cartesian product of the rational numbers ``n`` times; this is - ``QQ^n`` with the Hadamard product. + """ + n = M.nrows() + if M.ncols() != n: + raise ArgumentError("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]])) - * The Jordan spin algebra on ``QQ^n``. + # We can drop the imaginaries here. + return block_matrix(field.base_ring(), n, blocks) - * The ``n``-by-``n`` rational symmetric matrices with the symmetric - product. - Later this might be extended to return Cartesian products of the - EJAs above. +def _unembed_complex_matrix(M): + """ + The inverse of _embed_complex_matrix(). - TESTS:: + EXAMPLES:: - sage: random_eja() - Euclidean Jordan algebra of degree... + 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): + """ + 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. + """ + pass +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 + the reals. + """ + F = QuadraticField(-1, 'i') + i = F.gen() + S = _real_symmetric_basis(n, field=F) + T = [] + for s in S: + T.append(s) + T.append(i*s) + embed_T = [ _embed_complex_matrix(t) for t in T ] + Qs = _multiplication_table_from_matrix_basis(embed_T) + return FiniteDimensionalEuclideanJordanAlgebra(field, Qs, rank=n) + +def QuaternionHermitianSimpleEJA(n): + """ + 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. + """ + n = 3 + pass + +def JordanSpinSimpleEJA(n): + """ + 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. """ - n = ZZ.random_element(1,10).abs() - constructor = choice([eja_rn, eja_ln, eja_sn]) - return constructor(dimension=n, field=QQ) + pass