sage: RealCartesianProductEJA(3, prefix='r').gens()
(r0, r1, r2)
+ Our inner product satisfies the Jordan axiom::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(1,5)
+ sage: J = RealCartesianProductEJA(n)
+ 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
+
"""
def __init__(self, n, field=QQ, **kwargs):
V = VectorSpace(field, n)
-def _real_symmetric_basis(n, field=QQ):
+def _real_symmetric_basis(n, field):
"""
Return a basis for the space of real symmetric n-by-n matrices.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import _real_symmetric_basis
+
+ TESTS::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(1,5)
+ sage: B = _real_symmetric_basis(n, QQbar)
+ sage: all( M.is_symmetric() for M in B)
+ True
+
"""
# The basis of symmetric matrices, as matrices, in their R^(n-by-n)
# coordinates.
return tuple(S)
-def _complex_hermitian_basis(n, field=QQ):
+def _complex_hermitian_basis(n, field):
"""
Returns a basis for the space of complex Hermitian n-by-n matrices.
return tuple(S)
-def _quaternion_hermitian_basis(n, field=QQ):
+def _quaternion_hermitian_basis(n, field):
"""
Returns a basis for the space of quaternion Hermitian n-by-n matrices.
sage: RealSymmetricEJA(3, prefix='q').gens()
(q0, q1, q2, q3, q4, q5)
+ Our inner product satisfies the Jordan axiom::
+
+ 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: z = J.random_element()
+ sage: (x*y).inner_product(z) == y.inner_product(x*z)
+ True
+
"""
def __init__(self, n, field=QQ, **kwargs):
- S = _real_symmetric_basis(n, field=field)
+ S = _real_symmetric_basis(n, field)
Qs = _multiplication_table_from_matrix_basis(S)
fdeja = super(RealSymmetricEJA, self)
sage: ComplexHermitianEJA(2, prefix='z').gens()
(z0, z1, z2, z3)
+ Our inner product satisfies the Jordan axiom::
+
+ 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: z = J.random_element()
+ sage: (x*y).inner_product(z) == y.inner_product(x*z)
+ True
+
"""
def __init__(self, n, field=QQ, **kwargs):
- S = _complex_hermitian_basis(n)
+ S = _complex_hermitian_basis(n, field)
Qs = _multiplication_table_from_matrix_basis(S)
fdeja = super(ComplexHermitianEJA, self)
sage: QuaternionHermitianEJA(2, prefix='a').gens()
(a0, a1, a2, a3, a4, a5)
+ Our inner product satisfies the Jordan axiom::
+
+ 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: z = J.random_element()
+ sage: (x*y).inner_product(z) == y.inner_product(x*z)
+ True
+
"""
def __init__(self, n, field=QQ, **kwargs):
- S = _quaternion_hermitian_basis(n)
+ S = _quaternion_hermitian_basis(n, field)
Qs = _multiplication_table_from_matrix_basis(S)
fdeja = super(QuaternionHermitianEJA, self)
sage: JordanSpinEJA(2, prefix='B').gens()
(B0, B1)
+ Our inner product satisfies the Jordan axiom::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(1,5)
+ sage: J = JordanSpinEJA(n)
+ 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
+
"""
def __init__(self, n, field=QQ, **kwargs):
V = VectorSpace(field, n)