X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_algebra.py;h=28e86c25e3a3d315cf8fefb325fc05ef4b03b427;hb=78b54b562e1eedbb919dc442b2b3f6996440a1a2;hp=d06efb59b14c1c7ecf1fb3fda375168ded981e6e;hpb=634dab08d886610e41dfd363a0a608a9405abe8e;p=sage.d.git diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py index d06efb5..28e86c2 100644 --- a/mjo/eja/eja_algebra.py +++ b/mjo/eja/eja_algebra.py @@ -5,7 +5,6 @@ are used in optimization, and have some additional nice methods beyond 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 @@ -18,7 +17,6 @@ from sage.rings.number_field.number_field import QuadraticField 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 @@ -50,7 +48,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): """ 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) @@ -60,6 +58,15 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): 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): """ @@ -146,10 +153,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): 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): """ @@ -243,10 +247,11 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): # 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(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) @@ -374,6 +379,41 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): 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 matrix(self._multiplication_table) + + def natural_basis(self): """ Return a more-natural representation of this algebra's basis. @@ -437,7 +477,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): sage: J.one() e0 + e1 + e2 + e3 + e4 - TESTS:: + TESTS: The identity element acts like the identity:: @@ -597,16 +637,12 @@ class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra): """ 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) @@ -777,10 +813,7 @@ 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. 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 @@ -792,20 +825,14 @@ def _multiplication_table_from_matrix_basis(basis): 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): @@ -1264,24 +1291,27 @@ class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra): """ 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)