what can be supported in a general Jordan Algebra.
"""
-from sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra import FiniteDimensionalAlgebra
+#from sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra import FiniteDimensionalAlgebra
from sage.algebras.quatalg.quaternion_algebra import QuaternionAlgebra
from sage.categories.finite_dimensional_algebras_with_basis import FiniteDimensionalAlgebrasWithBasis
+from sage.combinat.free_module import CombinatorialFreeModule
from sage.matrix.constructor import matrix
from sage.misc.cachefunc import cached_method
from sage.misc.prandom import choice
from sage.structure.category_object import normalize_names
from mjo.eja.eja_element import FiniteDimensionalEuclideanJordanAlgebraElement
-from mjo.eja.eja_utils import _vec2mat, _mat2vec
-
-class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
- @staticmethod
- def __classcall_private__(cls,
- field,
- mult_table,
- rank,
- names='e',
- assume_associative=False,
- category=None,
- natural_basis=None):
- n = len(mult_table)
- mult_table = [b.base_extend(field) for b in mult_table]
- for b in mult_table:
- b.set_immutable()
- if not (is_Matrix(b) and b.dimensions() == (n, n)):
- raise ValueError("input is not a multiplication table")
- mult_table = tuple(mult_table)
-
- cat = FiniteDimensionalAlgebrasWithBasis(field)
- cat.or_subcategory(category)
- if assume_associative:
- cat = cat.Associative()
-
- names = normalize_names(n, names)
-
- fda = super(FiniteDimensionalEuclideanJordanAlgebra, cls)
- return fda.__classcall__(cls,
- field,
- mult_table,
- rank,
- assume_associative=assume_associative,
- names=names,
- category=cat,
- natural_basis=natural_basis)
-
+from mjo.eja.eja_utils import _mat2vec
+class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule):
def __init__(self,
field,
mult_table,
rank,
- names='e',
- assume_associative=False,
+ prefix='e',
category=None,
natural_basis=None):
"""
self._rank = rank
self._natural_basis = natural_basis
self._multiplication_table = mult_table
+ if category is None:
+ category = FiniteDimensionalAlgebrasWithBasis(field).Unital()
fda = super(FiniteDimensionalEuclideanJordanAlgebra, self)
fda.__init__(field,
- mult_table,
- names=names,
+ range(len(mult_table)),
+ prefix=prefix,
category=category)
+ self.print_options(bracket='')
def _repr_(self):
Euclidean Jordan algebra of degree 3 over Real Double Field
"""
+ # TODO: change this to say "dimension" and fix all the tests.
fmt = "Euclidean Jordan algebra of degree {} over {}"
- return fmt.format(self.degree(), self.base_ring())
+ return fmt.format(self.dimension(), self.base_ring())
+ def product_on_basis(self, i, j):
+ ei = self.basis()[i]
+ ej = self.basis()[j]
+ Lei = self._multiplication_table[i]
+ return self.from_vector(Lei*ej.to_vector())
def _a_regular_element(self):
"""
"""
z = self._a_regular_element()
V = self.vector_space()
- V1 = V.span_of_basis( (z**k).vector() for k in range(self.rank()) )
+ V1 = V.span_of_basis( (z**k).to_vector() for k in range(self.rank()) )
b = (V1.basis() + V1.complement().basis())
return V.span_of_basis(b)
n = self.dimension()
# Construct a new algebra over a multivariate polynomial ring...
- names = ['X' + str(i) for i in range(1,n+1)]
+ names = tuple('X' + str(i) for i in range(1,n+1))
R = PolynomialRing(self.base_ring(), names)
- J = FiniteDimensionalEuclideanJordanAlgebra(R,
- self._multiplication_table,
- r)
+ J = FiniteDimensionalEuclideanJordanAlgebra(
+ R,
+ tuple(self._multiplication_table),
+ r)
idmat = matrix.identity(J.base_ring(), n)
# Handle the zeroth power separately, because computing
# the unit element in J is mathematically suspect.
- x0 = W.coordinates(self.one().vector())
- l1 = [ matrix.column(x0) ]
- l1 += [ matrix.column(W.coordinates((x**k).vector()))
+ x0 = W.coordinate_vector(self.one().to_vector())
+ l1 = [ x0.column() ]
+ l1 += [ W.coordinate_vector((x**k).to_vector()).column()
for k in range(1,r) ]
l2 = [idmat.column(k-1).column() for k in range(r+1, n+1)]
A_of_x = matrix.block(R, 1, n, (l1 + l2))
- xr = W.coordinates((x**r).vector())
+ xr = W.coordinate_vector((x**r).to_vector())
return (A_of_x, x, xr, A_of_x.det())
sage: J = JordanSpinEJA(3)
sage: p = J.characteristic_polynomial(); p
X1^2 - X2^2 - X3^2 + (-2*t)*X1 + t^2
- sage: xvec = J.one().vector()
+ sage: xvec = J.one().to_vector()
sage: p(*xvec)
t^2 - 2*t + 1
sage: J = RealSymmetricEJA(2)
sage: J.basis()
- Family (e0, e1, e2)
+ Finite family {0: e0, 1: e1, 2: e2}
sage: J.natural_basis()
(
[1 0] [0 1] [0 0]
sage: J = JordanSpinEJA(2)
sage: J.basis()
- Family (e0, e1)
+ Finite family {0: e0, 1: e1}
sage: J.natural_basis()
(
[1] [0]
"""
if self._natural_basis is None:
- return tuple( b.vector().column() for b in self.basis() )
+ return tuple( b.to_vector().column() for b in self.basis() )
else:
return self._natural_basis
# Now if there's an identity element in the algebra, this should work.
coeffs = A.solve_right(b)
- return self.linear_combination(zip(coeffs,self.gens()))
+ return self.linear_combination(zip(self.gens(), coeffs))
def rank(self):
Vector space of dimension 3 over Rational Field
"""
- return self.zero().vector().parent().ambient_vector_space()
+ return self.zero().to_vector().parent().ambient_vector_space()
Element = FiniteDimensionalEuclideanJordanAlgebraElement
e2
"""
- @staticmethod
- def __classcall_private__(cls, n, field=QQ):
- # The FiniteDimensionalAlgebra constructor takes a list of
- # matrices, the ith representing right multiplication by the ith
- # basis element in the vector space. So if e_1 = (1,0,0), then
- # right (Hadamard) multiplication of x by e_1 picks out the first
+ def __init__(self, n, field=QQ):
+ # The superclass constructor takes a list of matrices, the ith
+ # representing right multiplication by the ith basis element
+ # in the vector space. So if e_1 = (1,0,0), then right
+ # (Hadamard) multiplication of x by e_1 picks out the first
# component of x; and likewise for the ith basis element e_i.
Qs = [ matrix(field, n, n, lambda k,j: 1*(k == j == i))
for i in xrange(n) ]
- fdeja = super(RealCartesianProductEJA, cls)
- return fdeja.__classcall_private__(cls, field, Qs, rank=n)
+ fdeja = super(RealCartesianProductEJA, self)
+ return fdeja.__init__(field, Qs, rank=n)
def inner_product(self, x, y):
return _usual_ip(x,y)
dimension = basis[0].nrows()
V = VectorSpace(field, dimension**2)
- 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() )
+ W = V.span_of_basis( _mat2vec(s) for s in basis )
Qs = []
- for s in S:
+ for s in basis:
# Brute force the multiplication-by-s matrix by looping
# through all elements of the basis and doing the computation
- # to find out what the corresponding row should be. BEWARE:
- # these multiplication tables won't be symmetric! It therefore
- # becomes REALLY IMPORTANT that the underlying algebra
- # constructor uses ROW vectors and not COLUMN vectors. That's
- # why we're computing rows here and not columns.
- Q_rows = []
- for t in S:
- this_row = _mat2vec((s*t + t*s)/2)
- Q_rows.append(W.coordinates(this_row))
- Q = matrix(field, W.dimension(), Q_rows)
+ # to find out what the corresponding row should be.
+ Q_cols = []
+ for t in basis:
+ this_col = _mat2vec((s*t + t*s)/2)
+ Q_cols.append(W.coordinates(this_col))
+ Q = matrix.column(field, W.dimension(), Q_cols)
Qs.append(Q)
- return (Qs, S)
+ return Qs
def _embed_complex_matrix(M):
# The usual inner product on R^n.
def _usual_ip(x,y):
- return x.vector().inner_product(y.vector())
+ return x.to_vector().inner_product(y.to_vector())
# The inner product used for the real symmetric simple EJA.
# We keep it as a separate function because e.g. the complex
True
"""
- @staticmethod
- def __classcall_private__(cls, n, field=QQ):
+ def __init__(self, n, field=QQ):
S = _real_symmetric_basis(n, field=field)
- (Qs, T) = _multiplication_table_from_matrix_basis(S)
+ Qs = _multiplication_table_from_matrix_basis(S)
- fdeja = super(RealSymmetricEJA, cls)
- return fdeja.__classcall_private__(cls,
- field,
- Qs,
- rank=n,
- natural_basis=T)
+ fdeja = super(RealSymmetricEJA, self)
+ return fdeja.__init__(field,
+ Qs,
+ rank=n,
+ natural_basis=S)
def inner_product(self, x, y):
return _matrix_ip(x,y)
True
"""
- @staticmethod
- def __classcall_private__(cls, n, field=QQ):
+ def __init__(self, n, field=QQ):
S = _complex_hermitian_basis(n)
- (Qs, T) = _multiplication_table_from_matrix_basis(S)
+ Qs = _multiplication_table_from_matrix_basis(S)
+
+ fdeja = super(ComplexHermitianEJA, self)
+ return fdeja.__init__(field,
+ Qs,
+ rank=n,
+ natural_basis=S)
- fdeja = super(ComplexHermitianEJA, cls)
- return fdeja.__classcall_private__(cls,
- field,
- Qs,
- rank=n,
- natural_basis=T)
def inner_product(self, x, y):
# Since a+bi on the diagonal is represented as
True
"""
- @staticmethod
- def __classcall_private__(cls, n, field=QQ):
+ def __init__(self, n, field=QQ):
S = _quaternion_hermitian_basis(n)
- (Qs, T) = _multiplication_table_from_matrix_basis(S)
+ Qs = _multiplication_table_from_matrix_basis(S)
- fdeja = super(QuaternionHermitianEJA, cls)
- return fdeja.__classcall_private__(cls,
- field,
- Qs,
- rank=n,
- natural_basis=T)
+ fdeja = super(QuaternionHermitianEJA, self)
+ return fdeja.__init__(field,
+ Qs,
+ rank=n,
+ natural_basis=S)
def inner_product(self, x, y):
# Since a+bi+cj+dk on the diagonal is represented as
0
"""
- @staticmethod
- def __classcall_private__(cls, n, field=QQ):
+ def __init__(self, n, field=QQ):
Qs = []
id_matrix = matrix.identity(field, n)
for i in xrange(n):
# 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).
- fdeja = super(JordanSpinEJA, cls)
- return fdeja.__classcall_private__(cls, field, Qs, rank=min(n,2))
+ fdeja = super(JordanSpinEJA, self)
+ return fdeja.__init__(field, Qs, rank=min(n,2))
def inner_product(self, x, y):
return _usual_ip(x,y)
projects / sage.d.git / blobdiff