+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)
+ 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.
+
+ 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]]``.
+
+ 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 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]
+
+ 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.
+
+ 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())
+
+# 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()
+
+
+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)
+ sage: J = RealSymmetricSimpleEJA(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 = RealSymmetricSimpleEJA(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)
+
+ return FiniteDimensionalEuclideanJordanAlgebra(field,
+ Qs,
+ rank=n,
+ natural_basis=T,
+ inner_product=_matrix_ip)
+
+
+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.
+
+ TESTS:
+
+ The degree of this algebra is `n^2`::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(1,5)
+ sage: J = ComplexHermitianSimpleEJA(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 = ComplexHermitianSimpleEJA(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)
+
+ # 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
+
+ return FiniteDimensionalEuclideanJordanAlgebra(field,
+ Qs,
+ rank=n,
+ natural_basis=T,
+ inner_product=ip)
+
+
+def QuaternionHermitianSimpleEJA(n, field=QQ):
+ """
+ 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.
+
+ TESTS:
+
+ The degree of this algebra is `n^2`::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(1,5)
+ sage: J = QuaternionHermitianSimpleEJA(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 = QuaternionHermitianSimpleEJA(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 = _quaternion_hermitian_basis(n)
+ (Qs, T) = _multiplication_table_from_matrix_basis(S)
+
+ # 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.
+ ip = lambda X,Y: _matrix_ip(X,Y)/4
+
+ return FiniteDimensionalEuclideanJordanAlgebra(field,
+ Qs,
+ rank=n,
+ natural_basis=T,
+ inner_product=ip)
+
+
+def OctonionHermitianSimpleEJA(n):
+ """
+ This shit be crazy. It has dimension 27 over the reals.
+ """
+ n = 3
+ pass