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.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 _mat2vec
"""
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)
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 = matrix(
+ [ map(lambda x: self.from_vector(x), ls)
+ for ls in mult_table ] )
+ self._multiplication_table.set_immutable()
+
+
+ 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):
"""
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())
+ return self._multiplication_table[i,j]
def _a_regular_element(self):
"""
# Construct a new algebra over a multivariate polynomial ring...
names = tuple('X' + str(i) for i in range(1,n+1))
R = PolynomialRing(self.base_ring(), names)
- J = FiniteDimensionalEuclideanJordanAlgebra(
- R,
- tuple(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([ self._multiplication_table[i,j].to_vector()
+ for j in range(self._multiplication_table.nrows()) ])
+ for i in range(self._multiplication_table.ncols()) ]
+ 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()))
+ x = J.from_vector(W(R.gens()))
# Handle the zeroth power separately, because computing
# the unit element in J is mathematically suspect.
return x.trace_inner_product(y)
+ def multiplication_table(self):
+ """
+ Return a readable matrix representation of this algebra's
+ multiplication table. The (i,j)th entry in the matrix contains
+ the product of the ith basis element with the jth.
+
+ This is not extraordinarily useful, but it overrides a superclass
+ method that would otherwise just crash and complain about the
+ algebra being infinite.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
+ ....: RealCartesianProductEJA)
+
+ EXAMPLES::
+
+ sage: J = RealCartesianProductEJA(3)
+ sage: J.multiplication_table()
+ [e0 0 0]
+ [ 0 e1 0]
+ [ 0 0 e2]
+
+ ::
+
+ sage: J = JordanSpinEJA(3)
+ sage: J.multiplication_table()
+ [e0 e1 e2]
+ [e1 e0 0]
+ [e2 0 e0]
+
+ """
+ return self._multiplication_table
+
+
def natural_basis(self):
"""
Return a more-natural representation of this algebra's basis.
sage: J.one()
e0 + e1 + e2 + e3 + e4
- TESTS::
+ TESTS:
The identity element acts like the identity::
"""
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) ]
+ 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, Qs, rank=n)
+ 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
V = VectorSpace(field, dimension**2)
W = V.span_of_basis( _mat2vec(s) for s in basis )
+ n = len(basis)
+ mult_table = [[W.zero() for j in range(n)] for i 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))
- Qs = []
- 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.
- 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
+ return mult_table
def _embed_complex_matrix(M):
"""
def __init__(self, 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)
+ V = VectorSpace(field, n)
+ mult_table = [[V.zero() for j in range(n)] for i in range(n)]
+ for i in range(n):
+ for j in range(n):
+ x = V.gen(i)
+ y = V.gen(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, self)
- return fdeja.__init__(field, Qs, rank=min(n,2))
+ 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