from sage.categories.magmatic_algebras import MagmaticAlgebras
from sage.combinat.free_module import CombinatorialFreeModule
from sage.matrix.constructor import matrix
+from sage.matrix.matrix_space import MatrixSpace
from sage.misc.cachefunc import cached_method
from sage.misc.prandom import choice
from sage.misc.table import table
-from sage.modules.free_module import VectorSpace
+from sage.modules.free_module import FreeModule, VectorSpace
from sage.rings.integer_ring import ZZ
from sage.rings.number_field.number_field import QuadraticField
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
determinant).
"""
z = self._a_regular_element()
- V = self.vector_space()
- V1 = V.span_of_basis( (z**k).to_vector() for k in range(self.rank()) )
+ # Don't use the parent vector space directly here in case this
+ # happens to be a subalgebra. In that case, we would be e.g.
+ # two-dimensional but span_of_basis() would expect three
+ # coordinates.
+ V = VectorSpace(self.base_ring(), self.vector_space().dimension())
+ basis = [ (z**k).to_vector() for k in range(self.rank()) ]
+ V1 = V.span_of_basis( basis )
b = (V1.basis() + V1.complement().basis())
return V.span_of_basis(b)
# have multivatiate polynomial entries.
names = tuple('X' + str(i) for i in range(1,n+1))
R = PolynomialRing(self.base_ring(), names)
- V = self.vector_space().change_ring(R)
+
+ # Using change_ring() on the parent's vector space doesn't work
+ # here because, in a subalgebra, that vector space has a basis
+ # and change_ring() tries to bring the basis along with it. And
+ # that doesn't work unless the new ring is a PID, which it usually
+ # won't be.
+ V = FreeModule(R,n)
# Now let x = (X1,X2,...,Xn) be the vector whose entries are
# indeterminates...
return x.trace_inner_product(y)
+ def is_trivial(self):
+ """
+ Return whether or not this algebra is trivial.
+
+ A trivial algebra contains only the zero element.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
+
+ EXAMPLES::
+
+ sage: J = ComplexHermitianEJA(3)
+ sage: J.is_trivial()
+ False
+ sage: A = J.zero().subalgebra_generated_by()
+ sage: A.is_trivial()
+ True
+
+ """
+ return self.dimension() == 0
+
+
def multiplication_table(self):
"""
Return a visual representation of this algebra's multiplication
"""
if self._natural_basis is None:
- return tuple( b.to_vector().column() for b in self.basis() )
+ M = self.natural_basis_space()
+ return tuple( M(b.to_vector()) for b in self.basis() )
else:
return self._natural_basis
+ def natural_basis_space(self):
+ """
+ Return the matrix space in which this algebra's natural basis
+ elements live.
+ """
+ if self._natural_basis is None or len(self._natural_basis) == 0:
+ return MatrixSpace(self.base_ring(), self.dimension(), 1)
+ else:
+ return self._natural_basis[0].matrix_space()
+
+
@cached_method
def one(self):
"""
return self.linear_combination(zip(self.gens(), coeffs))
+ def random_element(self):
+ # Temporary workaround for https://trac.sagemath.org/ticket/28327
+ if self.is_trivial():
+ return self.zero()
+ else:
+ s = super(FiniteDimensionalEuclideanJordanAlgebra, self)
+ return s.random_element()
+
+
def rank(self):
"""
Return the rank of this EJA.
sage: e2*e2
e2
+ TESTS:
+
+ We can change the generator prefix::
+
+ sage: RealCartesianProductEJA(3, prefix='r').gens()
+ (r0, r1, r2)
+
"""
- def __init__(self, n, field=QQ):
+ def __init__(self, n, field=QQ, **kwargs):
V = VectorSpace(field, n)
mult_table = [ [ V.gen(i)*(i == j) for j in range(n) ]
for i in range(n) ]
fdeja = super(RealCartesianProductEJA, self)
- return fdeja.__init__(field, mult_table, rank=n)
+ return fdeja.__init__(field, mult_table, rank=n, **kwargs)
def inner_product(self, x, y):
return _usual_ip(x,y)
sage: J(expected) == x*y
True
+ We can change the generator prefix::
+
+ sage: RealSymmetricEJA(3, prefix='q').gens()
+ (q0, q1, q2, q3, q4, q5)
+
"""
- def __init__(self, n, field=QQ):
+ def __init__(self, n, field=QQ, **kwargs):
S = _real_symmetric_basis(n, field=field)
Qs = _multiplication_table_from_matrix_basis(S)
return fdeja.__init__(field,
Qs,
rank=n,
- natural_basis=S)
+ natural_basis=S,
+ **kwargs)
def inner_product(self, x, y):
return _matrix_ip(x,y)
sage: J(expected) == x*y
True
+ We can change the generator prefix::
+
+ sage: ComplexHermitianEJA(2, prefix='z').gens()
+ (z0, z1, z2, z3)
+
"""
- def __init__(self, n, field=QQ):
+ def __init__(self, n, field=QQ, **kwargs):
S = _complex_hermitian_basis(n)
Qs = _multiplication_table_from_matrix_basis(S)
return fdeja.__init__(field,
Qs,
rank=n,
- natural_basis=S)
+ natural_basis=S,
+ **kwargs)
def inner_product(self, x, y):
sage: J(expected) == x*y
True
+ We can change the generator prefix::
+
+ sage: QuaternionHermitianEJA(2, prefix='a').gens()
+ (a0, a1, a2, a3, a4, a5)
+
"""
- def __init__(self, n, field=QQ):
+ def __init__(self, n, field=QQ, **kwargs):
S = _quaternion_hermitian_basis(n)
Qs = _multiplication_table_from_matrix_basis(S)
return fdeja.__init__(field,
Qs,
rank=n,
- natural_basis=S)
+ natural_basis=S,
+ **kwargs)
def inner_product(self, x, y):
# Since a+bi+cj+dk on the diagonal is represented as
sage: e2*e3
0
+ We can change the generator prefix::
+
+ sage: JordanSpinEJA(2, prefix='B').gens()
+ (B0, B1)
+
"""
- def __init__(self, n, field=QQ):
+ def __init__(self, n, field=QQ, **kwargs):
V = VectorSpace(field, n)
mult_table = [[V.zero() for j in range(n)] for i in range(n)]
for i in range(n):
# one-dimensional ambient space (because the rank is bounded by
# the ambient dimension).
fdeja = super(JordanSpinEJA, self)
- return fdeja.__init__(field, mult_table, rank=min(n,2))
+ return fdeja.__init__(field, mult_table, rank=min(n,2), **kwargs)
def inner_product(self, x, y):
return _usual_ip(x,y)
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