what can be supported in a general Jordan Algebra.
"""
-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.rings.polynomial.polynomial_ring_constructor import PolynomialRing
from sage.rings.rational_field import QQ
from sage.structure.element import is_Matrix
-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='')
+
+ # The multiplication table we're given is necessarily in terms
+ # of vectors, because we don't have an algebra yet for
+ # anything to be an element of. However, it's faster in the
+ # long run to have the multiplication table be in terms of
+ # algebra elements. We do this after calling the superclass
+ # constructor so that from_vector() knows what to do.
+ self._multiplication_table = [ map(lambda x: self.from_vector(x), ls)
+ for ls in mult_table ]
+
+
+ def _element_constructor_(self, elt):
+ """
+ Construct an element of this algebra from its natural
+ representation.
+
+ This gets called only after the parent element _call_ method
+ fails to find a coercion for the argument.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
+ ....: RealCartesianProductEJA,
+ ....: RealSymmetricEJA)
+
+ EXAMPLES:
+
+ The identity in `S^n` is converted to the identity in the EJA::
+
+ sage: J = RealSymmetricEJA(3)
+ sage: I = matrix.identity(QQ,3)
+ sage: J(I) == J.one()
+ True
+
+ This skew-symmetric matrix can't be represented in the EJA::
+
+ sage: J = RealSymmetricEJA(3)
+ sage: A = matrix(QQ,3, lambda i,j: i-j)
+ sage: J(A)
+ Traceback (most recent call last):
+ ...
+ ArithmeticError: vector is not in free module
+
+ TESTS:
+
+ Ensure that we can convert any element of the two non-matrix
+ simple algebras (whose natural representations are their usual
+ vector representations) back and forth faithfully::
+
+ sage: set_random_seed()
+ sage: J = RealCartesianProductEJA(5)
+ sage: x = J.random_element()
+ sage: J(x.to_vector().column()) == x
+ True
+ sage: J = JordanSpinEJA(5)
+ sage: x = J.random_element()
+ sage: J(x.to_vector().column()) == x
+ True
+
+ """
+ natural_basis = self.natural_basis()
+ if elt not in natural_basis[0].matrix_space():
+ raise ValueError("not a naturally-represented algebra element")
+
+ # Thanks for nothing! Matrix spaces aren't vector
+ # spaces in Sage, so we have to figure out its
+ # natural-basis coordinates ourselves.
+ V = VectorSpace(elt.base_ring(), elt.nrows()*elt.ncols())
+ W = V.span_of_basis( _mat2vec(s) for s in natural_basis )
+ coords = W.coordinate_vector(_mat2vec(elt))
+ return self.from_vector(coords)
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):
+ return self._multiplication_table[i][j]
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)
+ # Hack around the fact that our multiplication table is in terms of
+ # algebra elements but the constructor wants it in terms of vectors.
+ vmt = [ tuple(map(lambda x: x.to_vector(), ls))
+ for ls in self._multiplication_table ]
+ J = FiniteDimensionalEuclideanJordanAlgebra(R, tuple(vmt), r)
idmat = matrix.identity(J.base_ring(), n)
# We want the middle equivalent thing in our matrix, but use
# the first equivalent thing instead so that we can pass in
# standard coordinates.
- x = J(W(R.gens()))
- l1 = [matrix.column(W.coordinates((x**k).vector())) for k in range(r)]
+ x = J.from_vector(W(R.gens()))
+
+ # Handle the zeroth power separately, because computing
+ # the unit element in J is mathematically suspect.
+ 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
+ @cached_method
+ def one(self):
+ """
+ Return the unit element of this algebra.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (RealCartesianProductEJA,
+ ....: random_eja)
+
+ EXAMPLES::
+
+ sage: J = RealCartesianProductEJA(5)
+ sage: J.one()
+ e0 + e1 + e2 + e3 + e4
+
+ TESTS::
+
+ The identity element acts like the identity::
+
+ sage: set_random_seed()
+ sage: J = random_eja()
+ sage: x = J.random_element()
+ sage: J.one()*x == x and x*J.one() == x
+ True
+
+ The matrix of the unit element's operator is the identity::
+
+ sage: set_random_seed()
+ sage: J = random_eja()
+ sage: actual = J.one().operator().matrix()
+ sage: expected = matrix.identity(J.base_ring(), J.dimension())
+ sage: actual == expected
+ True
+
+ """
+ # We can brute-force compute the matrices of the operators
+ # that correspond to the basis elements of this algebra.
+ # If some linear combination of those basis elements is the
+ # algebra identity, then the same linear combination of
+ # their matrices has to be the identity matrix.
+ #
+ # Of course, matrices aren't vectors in sage, so we have to
+ # appeal to the "long vectors" isometry.
+ oper_vecs = [ _mat2vec(g.operator().matrix()) for g in self.gens() ]
+
+ # Now we use basis linear algebra to find the coefficients,
+ # of the matrices-as-vectors-linear-combination, which should
+ # work for the original algebra basis too.
+ A = matrix.column(self.base_ring(), oper_vecs)
+
+ # We used the isometry on the left-hand side already, but we
+ # still need to do it for the right-hand side. Recall that we
+ # wanted something that summed to the identity matrix.
+ b = _mat2vec( matrix.identity(self.base_ring(), self.dimension()) )
+
+ # Now if there's an identity element in the algebra, this should work.
+ coeffs = A.solve_right(b)
+ return self.linear_combination(zip(self.gens(), coeffs))
+
+
def rank(self):
"""
Return the rank of this EJA.
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
- # 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)
+ def __init__(self, n, field=QQ):
+ V = VectorSpace(field, n)
+ mult_table = [ [ V.basis()[i]*(i == j) for i in range(n) ]
+ for j in range(n) ]
+
+ fdeja = super(RealCartesianProductEJA, self)
+ return fdeja.__init__(field, mult_table, rank=n)
def inner_product(self, x, y):
return _usual_ip(x,y)
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. 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.
+ elements) for an algebra of those matrices.
"""
# In S^2, for example, we nominally have four coordinates even
# though the space is of dimension three only. The vector space V
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() )
-
- Qs = []
- for s in S:
- # 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)
- Qs.append(Q)
-
- return (Qs, S)
+ W = V.span_of_basis( _mat2vec(s) for s in basis )
+ n = len(basis)
+ mult_table = [[W.zero() for i in range(n)] for j in range(n)]
+ for i in range(n):
+ for j in range(n):
+ mat_entry = (basis[i]*basis[j] + basis[j]*basis[i])/2
+ mult_table[i][j] = W.coordinate_vector(_mat2vec(mat_entry))
+
+ return mult_table
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
TESTS:
- The degree of this algebra is `(n^2 + n) / 2`::
+ The dimension of this algebra is `(n^2 + n) / 2`::
sage: set_random_seed()
sage: n = ZZ.random_element(1,5)
sage: J = RealSymmetricEJA(n)
- sage: J.degree() == (n^2 + n)/2
+ sage: J.dimension() == (n^2 + n)/2
True
The Jordan multiplication is what we think it is::
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)
TESTS:
- The degree of this algebra is `n^2`::
+ The dimension of this algebra is `n^2`::
sage: set_random_seed()
sage: n = ZZ.random_element(1,5)
sage: J = ComplexHermitianEJA(n)
- sage: J.degree() == n^2
+ sage: J.dimension() == n^2
True
The Jordan multiplication is what we think it is::
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
TESTS:
- The degree of this algebra is `n^2`::
+ The dimension 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
+ sage: J.dimension() == 2*(n^2) - n
True
The Jordan multiplication is what we think it is::
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):
- Qs = []
- id_matrix = matrix.identity(field, n)
- for i in xrange(n):
- ei = id_matrix.column(i)
- Qi = matrix.zero(field, n)
- Qi.set_row(0, ei)
- Qi.set_column(0, ei)
- Qi += matrix.diagonal(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)
+ def __init__(self, n, field=QQ):
+ V = VectorSpace(field, n)
+ mult_table = [[V.zero() for i in range(n)] for j in range(n)]
+ for i in range(n):
+ for j in range(n):
+ x = V.basis()[i]
+ y = V.basis()[j]
+ x0 = x[0]
+ xbar = x[1:]
+ y0 = y[0]
+ ybar = y[1:]
+ # z = x*y
+ z0 = x.inner_product(y)
+ zbar = y0*xbar + x0*ybar
+ z = V([z0] + zbar.list())
+ mult_table[i][j] = z
# 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, mult_table, rank=min(n,2))
def inner_product(self, x, y):
return _usual_ip(x,y)
projects / sage.d.git / blobdiff