names='e',
assume_associative=False,
category=None,
- rank=None):
+ rank=None,
+ natural_basis=None):
n = len(mult_table)
mult_table = [b.base_extend(field) for b in mult_table]
for b in mult_table:
assume_associative=assume_associative,
names=names,
category=cat,
- rank=rank)
+ rank=rank,
+ natural_basis=natural_basis)
def __init__(self, field,
names='e',
assume_associative=False,
category=None,
- rank=None):
+ rank=None,
+ natural_basis=None):
"""
EXAMPLES:
"""
self._rank = rank
+ self._natural_basis = natural_basis
fda = super(FiniteDimensionalEuclideanJordanAlgebra, self)
fda.__init__(field,
mult_table,
fmt = "Euclidean Jordan algebra of degree {} over {}"
return fmt.format(self.degree(), self.base_ring())
+
+ def natural_basis(self):
+ """
+ Return a more-natural representation of this algebra's basis.
+
+ Every finite-dimensional Euclidean Jordan Algebra is a direct
+ sum of five simple algebras, four of which comprise Hermitian
+ matrices. This method returns the original "natural" basis
+ for our underlying vector space. (Typically, the natural basis
+ is used to construct the multiplication table in the first place.)
+
+ Note that this will always return a matrix. The standard basis
+ in `R^n` will be returned as `n`-by-`1` column matrices.
+
+ EXAMPLES::
+
+ sage: J = RealSymmetricSimpleEJA(2)
+ sage: J.basis()
+ Family (e0, e1, e2)
+ sage: J.natural_basis()
+ (
+ [1 0] [0 1] [0 0]
+ [0 0], [1 0], [0 1]
+ )
+
+ ::
+
+ sage: J = JordanSpinSimpleEJA(2)
+ sage: J.basis()
+ Family (e0, e1)
+ sage: J.natural_basis()
+ (
+ [1] [0]
+ [0], [1]
+ )
+
+ """
+ if self._natural_basis is None:
+ return tuple( b.vector().column() for b in self.basis() )
+ else:
+ return self._natural_basis
+
+
def rank(self):
"""
Return the rank of this EJA.
return self.span_of_powers().dimension()
- def matrix(self):
+ def operator_matrix(self):
"""
Return the matrix that represents left- (or right-)
multiplication by this element in the parent algebra.
fda_elt = FiniteDimensionalAlgebraElement(self.parent(), self)
return fda_elt.matrix().transpose()
- #
- # The plan is to eventually phase out "matrix()", which sounds
- # too much like "matrix_representation()", in favor of the more-
- # accurate "operator_matrix()". But we need to override matrix()
- # to keep parent class methods happy in the meantime.
- #
- operator_matrix = matrix
def minimal_polynomial(self):
# Beware, orthogonal but not normalized!
Sij = Eij + Eij.transpose()
S.append(Sij)
- return S
+ return tuple(S)
def _complex_hermitian_basis(n, field=QQ):
S.append(Sij_real)
Sij_imag = _embed_complex_matrix(I*Eij - I*Eij.transpose())
S.append(Sij_imag)
- return S
+ return tuple(S)
def _multiplication_table_from_matrix_basis(basis):
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.
+ 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
# 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 = [ vec2mat(b) for b in W.basis() ]
+ S = tuple( vec2mat(b) for b in W.basis() )
Qs = []
for s in S:
Q = matrix(field, W.dimension(), Q_rows)
Qs.append(Q)
- return Qs
+ return (Qs, S)
def _embed_complex_matrix(M):
"""
S = _real_symmetric_basis(n, field=field)
- Qs = _multiplication_table_from_matrix_basis(S)
+ (Qs, T) = _multiplication_table_from_matrix_basis(S)
- return FiniteDimensionalEuclideanJordanAlgebra(field,Qs,rank=n)
+ return FiniteDimensionalEuclideanJordanAlgebra(field,
+ Qs,
+ rank=n,
+ natural_basis=T)
def ComplexHermitianSimpleEJA(n, field=QQ):
"""
S = _complex_hermitian_basis(n)
- Qs = _multiplication_table_from_matrix_basis(S)
- return FiniteDimensionalEuclideanJordanAlgebra(field, Qs, rank=n)
+ (Qs, T) = _multiplication_table_from_matrix_basis(S)
+ return FiniteDimensionalEuclideanJordanAlgebra(field,
+ Qs,
+ rank=n,
+ natural_basis=T)
def QuaternionHermitianSimpleEJA(n):
projects / sage.d.git / blobdiff