+ n = ZZ.random_element(1,5)
+ constructor = choice([eja_rn,
+ JordanSpinSimpleEJA,
+ RealSymmetricSimpleEJA,
+ 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(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.
+
+ 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 _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]
+
+ """
+ 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::
+
+ 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 real submatrix')
+ if submat[0,1] != -submat[1,0]:
+ raise ValueError('bad imag submatrix')
+ z = submat[0,0] + submat[0,1]*i
+ elements.append(z)
+
+ return matrix(F, n/2, 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
+
+ """
+ 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
+
+ """
+ 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):
+ """
+ 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, 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 =
+ (<x_bar,y_bar>, x0*y_bar + y0*x_bar)``. It has dimension `n` over
+ the reals.