The inner product associated with this Euclidean Jordan algebra.
Will default to the trace inner product if nothing else.
+
+ EXAMPLES:
+
+ The inner product must satisfy its axiom for this algebra to truly
+ be a Euclidean Jordan Algebra::
+
+ sage: set_random_seed()
+ sage: J = random_eja()
+ sage: x = J.random_element()
+ sage: y = J.random_element()
+ sage: z = J.random_element()
+ sage: (x*y).inner_product(z) == y.inner_product(x*z)
+ True
+
"""
if (not x in self) or (not y in self):
raise TypeError("arguments must live in this algebra")
An element of a Euclidean Jordan algebra.
"""
+ def __init__(self, A, elt=None):
+ """
+ EXAMPLES:
+
+ The identity in `S^n` is converted to the identity in the EJA::
+
+ sage: J = RealSymmetricSimpleEJA(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: A = matrix(QQ,3, lambda i,j: i-j)
+ sage: J(A)
+ Traceback (most recent call last):
+ ...
+ ArithmeticError: vector is not in free module
+
+ """
+ # Goal: if we're given a matrix, and if it lives in our
+ # parent algebra's "natural ambient space," convert it
+ # into an algebra element.
+ #
+ # The catch is, we make a recursive call after converting
+ # the given matrix into a vector that lives in the algebra.
+ # This we need to try the parent class initializer first,
+ # to avoid recursing forever if we're given something that
+ # already fits into the algebra, but also happens to live
+ # in the parent's "natural ambient space" (this happens with
+ # vectors in R^n).
+ try:
+ FiniteDimensionalAlgebraElement.__init__(self, A, elt)
+ except ValueError:
+ natural_basis = A.natural_basis()
+ if elt in natural_basis[0].matrix_space():
+ # Thanks for nothing! Matrix spaces aren't vector
+ # spaces in Sage, so we have to figure out its
+ # natural-basis coordinates ourselves.
+ V = VectorSpace(elt.base_ring(), elt.nrows()**2)
+ W = V.span( _mat2vec(s) for s in natural_basis )
+ coords = W.coordinates(_mat2vec(elt))
+ FiniteDimensionalAlgebraElement.__init__(self, A, coords)
+
def __pow__(self, n):
"""
Return ``self`` raised to the power ``n``.
sage: J(x).inner_product(J(y))
32
+ The inner product on `S^n` is `<X,Y> = trace(X*Y)`, where
+ multiplication is the usual matrix multiplication in `S^n`,
+ so the inner product of the identity matrix with itself
+ should be the `n`::
+
+ sage: J = RealSymmetricSimpleEJA(3)
+ sage: J.one().inner_product(J.one())
+ 3
+
+ Likewise, the inner product on `C^n` is `<X,Y> =
+ Re(trace(X*Y))`, where we must necessarily take the real
+ part because the product of Hermitian matrices may not be
+ Hermitian::
+
+ sage: J = ComplexHermitianSimpleEJA(3)
+ sage: J.one().inner_product(J.one())
+ 3
+
+ Ditto for the quaternions::
+
+ sage: J = QuaternionHermitianSimpleEJA(3)
+ sage: J.one().inner_product(J.one())
+ 3
+
TESTS:
Ensure that we can always compute an inner product, and that
[0 0 0 0 1 0]
[0 0 0 0 0 1]
+ ::
+
+ sage: J = QuaternionHermitianSimpleEJA(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()
Qs = [ matrix(field, dimension, dimension, lambda k,j: 1*(k == j == i))
for i in xrange(dimension) ]
- # The usual inner product on R^n.
- ip = lambda x, y: x.vector().inner_product(y.vector())
-
return FiniteDimensionalEuclideanJordanAlgebra(field,
Qs,
rank=dimension,
- inner_product=ip)
+ inner_product=_usual_ip)
* 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.
constructor = choice([eja_rn,
JordanSpinSimpleEJA,
RealSymmetricSimpleEJA,
- ComplexHermitianSimpleEJA])
+ ComplexHermitianSimpleEJA,
+ QuaternionHermitianSimpleEJA])
return constructor(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())
+
+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
field = basis[0].base_ring()
dimension = basis[0].nrows()
- def mat2vec(m):
- return vector(field, m.list())
-
- def vec2mat(v):
- return matrix(field, dimension, v.list())
-
V = VectorSpace(field, dimension**2)
- W = V.span( mat2vec(s) for s in basis )
+ 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() )
+ S = tuple( _vec2mat(b) for b in W.basis() )
Qs = []
for s in S:
# why we're computing rows here and not columns.
Q_rows = []
for t in S:
- this_row = mat2vec((s*t + t*s)/2)
+ 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)
sage: x2 = F(1 + 2*i)
sage: x3 = F(-i)
sage: x4 = F(6)
- sage: M = matrix(F,2,[x1,x2,x3,x4])
+ sage: M = matrix(F,2,[[x1,x2],[x3,x4]])
sage: _embed_complex_matrix(M)
- [ 4 2| 1 -2]
- [-2 4| 2 1]
+ [ 4 -2| 1 2]
+ [ 2 4|-2 1]
[-----+-----]
- [ 0 1| 6 0]
- [-1 0| 0 6]
+ [ 0 -1| 6 0]
+ [ 1 0| 0 6]
"""
n = M.nrows()
for z in M.list():
a = z.real()
b = z.imag()
- blocks.append(matrix(field, 2, [[a,-b],[b,a]]))
+ blocks.append(matrix(field, 2, [[a,b],[-b,a]]))
# We can drop the imaginaries here.
return block_matrix(field.base_ring(), n, blocks)
....: [ 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]
+ [ 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:
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')
+ raise ValueError('bad on-diagonal submatrix')
if submat[0,1] != -submat[1,0]:
- raise ArgumentError('bad imag submatrix')
- z = submat[0,0] + submat[1,0]*i
+ 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]
+
+ """
+ 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::
+
+ 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
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)
+ natural_basis=T,
+ inner_product=_matrix_ip)
def ComplexHermitianSimpleEJA(n, field=QQ):
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)
+ natural_basis=T,
+ inner_product=ip)
-def QuaternionHermitianSimpleEJA(n):
+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
+
"""
- pass
+ 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):
"""
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)
Qi[0,0] = Qi[0,0] * ~field(2)
Qs.append(Qi)
- # The usual inner product on R^n.
- ip = lambda x, y: x.vector().inner_product(y.vector())
-
# 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=ip)
+ inner_product=_usual_ip)
projects / sage.d.git / blobdiff