inner_product=inner_product)
- def __init__(self, field,
+ def __init__(self,
+ field,
mult_table,
names='e',
assume_associative=False,
EXAMPLES::
- sage: J = RealSymmetricSimpleEJA(2)
+ sage: J = RealSymmetricEJA(2)
sage: J.basis()
Family (e0, e1, e2)
sage: J.natural_basis()
::
- sage: J = JordanSpinSimpleEJA(2)
+ sage: J = JordanSpinEJA(2)
sage: J.basis()
Family (e0, e1)
sage: J.natural_basis()
The identity in `S^n` is converted to the identity in the EJA::
- sage: J = RealSymmetricSimpleEJA(3)
+ sage: J = RealSymmetricEJA(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: J = RealSymmetricEJA(3)
sage: A = matrix(QQ,3, lambda i,j: i-j)
sage: J(A)
Traceback (most recent call last):
inner product on `R^n` (this example only works because the
basis for the Jordan algebra is the standard basis in `R^n`)::
- sage: J = JordanSpinSimpleEJA(3)
+ sage: J = JordanSpinEJA(3)
sage: x = vector(QQ,[1,2,3])
sage: y = vector(QQ,[4,5,6])
sage: x.inner_product(y)
so the inner product of the identity matrix with itself
should be the `n`::
- sage: J = RealSymmetricSimpleEJA(3)
+ sage: J = RealSymmetricEJA(3)
sage: J.one().inner_product(J.one())
3
part because the product of Hermitian matrices may not be
Hermitian::
- sage: J = ComplexHermitianSimpleEJA(3)
+ sage: J = ComplexHermitianEJA(3)
+ sage: J.one().inner_product(J.one())
+ 3
+
+ Ditto for the quaternions::
+
+ sage: J = QuaternionHermitianEJA(3)
sage: J.one().inner_product(J.one())
3
EXAMPLES::
- sage: J = JordanSpinSimpleEJA(2)
+ sage: J = JordanSpinEJA(2)
sage: e0,e1 = J.gens()
sage: x = e0 + e1
sage: x.det()
0
- sage: J = JordanSpinSimpleEJA(3)
+ sage: J = JordanSpinEJA(3)
sage: e0,e1,e2 = J.gens()
sage: x = e0 + e1 + e2
sage: x.det()
sage: set_random_seed()
sage: n = ZZ.random_element(1,10)
- sage: J = JordanSpinSimpleEJA(n)
+ sage: J = JordanSpinEJA(n)
sage: x = J.random_element()
sage: while x.is_zero():
....: x = J.random_element()
The identity element always has degree one, but any element
linearly-independent from it is regular::
- sage: J = JordanSpinSimpleEJA(5)
+ sage: J = JordanSpinEJA(5)
sage: J.one().is_regular()
False
sage: e0, e1, e2, e3, e4 = J.gens() # e0 is the identity
EXAMPLES::
- sage: J = JordanSpinSimpleEJA(4)
+ sage: J = JordanSpinEJA(4)
sage: J.one().degree()
1
sage: e0,e1,e2,e3 = J.gens()
sage: set_random_seed()
sage: n = ZZ.random_element(1,10)
- sage: J = JordanSpinSimpleEJA(n)
+ sage: J = JordanSpinEJA(n)
sage: x = J.random_element()
sage: x == x.coefficient(0)*J.one() or x.degree() == 2
True
sage: set_random_seed()
sage: n = ZZ.random_element(2,10)
- sage: J = JordanSpinSimpleEJA(n)
+ sage: J = JordanSpinEJA(n)
sage: y = J.random_element()
sage: while y == y.coefficient(0)*J.one():
....: y = J.random_element()
EXAMPLES::
- sage: J = ComplexHermitianSimpleEJA(3)
+ sage: J = ComplexHermitianEJA(3)
sage: J.one()
e0 + e5 + e8
sage: J.one().natural_representation()
[0 0 0 0 1 0]
[0 0 0 0 0 1]
+ ::
+
+ sage: J = QuaternionHermitianEJA(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()
sage: set_random_seed()
sage: n = ZZ.random_element(1,10)
- sage: J = JordanSpinSimpleEJA(n)
+ sage: J = JordanSpinEJA(n)
sage: x = J.random_element()
sage: x_vec = x.vector()
sage: x0 = x_vec[0]
sage: c = J.random_element().subalgebra_idempotent()
sage: c^2 == c
True
- sage: J = JordanSpinSimpleEJA(5)
+ sage: J = JordanSpinEJA(5)
sage: c = J.random_element().subalgebra_idempotent()
sage: c^2 == c
True
EXAMPLES::
- sage: J = JordanSpinSimpleEJA(3)
+ sage: J = JordanSpinEJA(3)
sage: e0,e1,e2 = J.gens()
sage: x = e0 + e1 + e2
sage: x.trace()
* 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.
"""
n = ZZ.random_element(1,5)
constructor = choice([eja_rn,
- JordanSpinSimpleEJA,
- RealSymmetricSimpleEJA,
- ComplexHermitianSimpleEJA])
+ JordanSpinEJA,
+ RealSymmetricEJA,
+ ComplexHermitianEJA,
+ QuaternionHermitianEJA])
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())
[ 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:
[ 2*i + 1 4*i + 3]
[ 10*i + 9 12*i + 11]
- TESTS::
+ TESTS:
+
+ Unembedding is the inverse of embedding::
sage: set_random_seed()
sage: F = QuadraticField(-1, 'i')
[-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()
sage: _unembed_quaternion_matrix(M)
[1 + 2*i + 3*j + 4*k]
- TESTS::
+ TESTS:
+
+ Unembedding is the inverse of embedding::
sage: set_random_seed()
sage: Q = QuaternionAlgebra(QQ, -1, -1)
return (X_mat*Y_mat).trace()
-def RealSymmetricSimpleEJA(n, field=QQ):
+class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra):
"""
The rank-n simple EJA consisting of real symmetric n-by-n
matrices, the usual symmetric Jordan product, and the trace inner
EXAMPLES::
- sage: J = RealSymmetricSimpleEJA(2)
+ sage: J = RealSymmetricEJA(2)
sage: e0, e1, e2 = J.gens()
sage: e0*e0
e0
sage: set_random_seed()
sage: n = ZZ.random_element(1,5)
- sage: J = RealSymmetricSimpleEJA(n)
+ sage: J = RealSymmetricEJA(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 = RealSymmetricEJA(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)
+ @staticmethod
+ def __classcall_private__(cls, n, field=QQ):
+ 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)
+ fdeja = super(RealSymmetricEJA, cls)
+ return fdeja.__classcall_private__(cls,
+ field,
+ Qs,
+ rank=n,
+ natural_basis=T)
+
+ def inner_product(self, x, y):
+ return _matrix_ip(x,y)
-def ComplexHermitianSimpleEJA(n, field=QQ):
+class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra):
"""
The rank-n simple EJA consisting of complex Hermitian n-by-n
matrices over the real numbers, the usual symmetric Jordan product,
sage: set_random_seed()
sage: n = ZZ.random_element(1,5)
- sage: J = ComplexHermitianSimpleEJA(n)
+ sage: J = ComplexHermitianEJA(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 = ComplexHermitianEJA(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)
+ @staticmethod
+ def __classcall_private__(cls, n, field=QQ):
+ 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
+ fdeja = super(ComplexHermitianEJA, cls)
+ return fdeja.__classcall_private__(cls,
+ field,
+ Qs,
+ rank=n,
+ natural_basis=T)
- return FiniteDimensionalEuclideanJordanAlgebra(field,
- Qs,
- rank=n,
- natural_basis=T,
- inner_product=ip)
+ def inner_product(self, x, y):
+ # 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.
+ return _matrix_ip(x,y)/2
-def QuaternionHermitianSimpleEJA(n):
+class QuaternionHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra):
"""
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.
+ TESTS:
+
+ The degree of this algebra is `n^2`::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(1,5)
+ sage: J = QuaternionHermitianEJA(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 = QuaternionHermitianEJA(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
+
"""
- n = 3
- pass
+ @staticmethod
+ def __classcall_private__(cls, n, field=QQ):
+ S = _quaternion_hermitian_basis(n)
+ (Qs, T) = _multiplication_table_from_matrix_basis(S)
-def JordanSpinSimpleEJA(n, field=QQ):
+ fdeja = super(QuaternionHermitianEJA, cls)
+ return fdeja.__classcall_private__(cls,
+ field,
+ Qs,
+ rank=n,
+ natural_basis=T)
+
+ def inner_product(self, x, y):
+ # 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.
+ return _matrix_ip(x,y)/4
+
+
+class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra):
"""
The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
with the usual inner product and jordan product ``x*y =
This multiplication table can be verified by hand::
- sage: J = JordanSpinSimpleEJA(4)
+ sage: J = JordanSpinEJA(4)
sage: e0,e1,e2,e3 = J.gens()
sage: e0*e0
e0
sage: e2*e3
0
- In one dimension, this is the reals under multiplication::
+ """
+ @staticmethod
+ def __classcall_private__(cls, n, field=QQ):
+ 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)
+
+ fdeja = super(JordanSpinEJA, cls)
+ return fdeja.__classcall_private__(cls, field, Qs)
- sage: J1 = JordanSpinSimpleEJA(1)
- sage: J2 = eja_rn(1)
- sage: J1 == J2
- True
+ def rank(self):
+ """
+ Return the rank of this Jordan Spin Algebra.
- """
- 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),
- inner_product=_usual_ip)
+ The rank of the spin algebra is two, unless we're in a
+ one-dimensional ambient space (because the rank is bounded by
+ the ambient dimension).
+ """
+ return min(self.dimension(),2)
+
+ def inner_product(self, x, y):
+ return _usual_ip(x,y)
projects / sage.d.git / blobdiff