X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_algebra.py;h=47d247be636c01d59a06a4f48da4d56b63b0f508;hb=801e9686b8eff405e50d9fb5cbf6f3b5a7c61117;hp=06f6f531ac46305da59ec2b4290ba808602cea50;hpb=0fd07263cc543e345f3cd7668938f8a0de70641f;p=sage.d.git diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py index 06f6f53..47d247b 100644 --- a/mjo/eja/eja_algebra.py +++ b/mjo/eja/eja_algebra.py @@ -9,6 +9,7 @@ from sage.algebras.quatalg.quaternion_algebra import QuaternionAlgebra from sage.categories.magmatic_algebras import MagmaticAlgebras from sage.combinat.free_module import CombinatorialFreeModule from sage.matrix.constructor import matrix +from sage.matrix.matrix_space import MatrixSpace from sage.misc.cachefunc import cached_method from sage.misc.prandom import choice from sage.misc.table import table @@ -134,13 +135,17 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): return self.zero() natural_basis = self.natural_basis() - if elt not in natural_basis[0].matrix_space(): + basis_space = natural_basis[0].matrix_space() + if elt not in basis_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()) + # Thanks for nothing! Matrix spaces aren't vector spaces in + # Sage, so we have to figure out its natural-basis coordinates + # ourselves. We use the basis space's ring instead of the + # element's ring because the basis space might be an algebraic + # closure whereas the base ring of the 3-by-3 identity matrix + # could be QQ instead of QQbar. + V = VectorSpace(basis_space.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) @@ -514,11 +519,23 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): """ if self._natural_basis is None: - return tuple( b.to_vector().column() for b in self.basis() ) + M = self.natural_basis_space() + return tuple( M(b.to_vector()) for b in self.basis() ) else: return self._natural_basis + def natural_basis_space(self): + """ + Return the matrix space in which this algebra's natural basis + elements live. + """ + if self._natural_basis is None or len(self._natural_basis) == 0: + return MatrixSpace(self.base_ring(), self.dimension(), 1) + else: + return self._natural_basis[0].matrix_space() + + @cached_method def one(self): """ @@ -709,6 +726,17 @@ class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra): sage: RealCartesianProductEJA(3, prefix='r').gens() (r0, r1, r2) + Our inner product satisfies the Jordan axiom:: + + sage: set_random_seed() + sage: n = ZZ.random_element(1,5) + sage: J = RealCartesianProductEJA(n) + sage: x = J.random_element() + sage: y = J.random_element() + sage: z = J.random_element() + sage: (x*y).inner_product(z) == y.inner_product(x*z) + True + """ def __init__(self, n, field=QQ, **kwargs): V = VectorSpace(field, n) @@ -773,9 +801,22 @@ def random_eja(): -def _real_symmetric_basis(n, field=QQ): +def _real_symmetric_basis(n, field): """ Return a basis for the space of real symmetric n-by-n matrices. + + SETUP:: + + sage: from mjo.eja.eja_algebra import _real_symmetric_basis + + TESTS:: + + sage: set_random_seed() + sage: n = ZZ.random_element(1,5) + sage: B = _real_symmetric_basis(n, QQbar) + sage: all( M.is_symmetric() for M in B) + True + """ # The basis of symmetric matrices, as matrices, in their R^(n-by-n) # coordinates. @@ -792,7 +833,7 @@ def _real_symmetric_basis(n, field=QQ): return tuple(S) -def _complex_hermitian_basis(n, field=QQ): +def _complex_hermitian_basis(n, field): """ Returns a basis for the space of complex Hermitian n-by-n matrices. @@ -804,7 +845,8 @@ def _complex_hermitian_basis(n, field=QQ): sage: set_random_seed() sage: n = ZZ.random_element(1,5) - sage: all( M.is_symmetric() for M in _complex_hermitian_basis(n) ) + sage: B = _complex_hermitian_basis(n, QQ) + sage: all( M.is_symmetric() for M in B) True """ @@ -833,7 +875,7 @@ def _complex_hermitian_basis(n, field=QQ): return tuple(S) -def _quaternion_hermitian_basis(n, field=QQ): +def _quaternion_hermitian_basis(n, field): """ Returns a basis for the space of quaternion Hermitian n-by-n matrices. @@ -845,7 +887,8 @@ def _quaternion_hermitian_basis(n, field=QQ): sage: set_random_seed() sage: n = ZZ.random_element(1,5) - sage: all( M.is_symmetric() for M in _quaternion_hermitian_basis(n) ) + sage: B = _quaternion_hermitian_basis(n, QQbar) + sage: all( M.is_symmetric() for M in B ) True """ @@ -1200,9 +1243,20 @@ class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra): sage: RealSymmetricEJA(3, prefix='q').gens() (q0, q1, q2, q3, q4, q5) + Our inner product satisfies the Jordan axiom:: + + sage: set_random_seed() + sage: n = ZZ.random_element(1,5) + sage: J = RealSymmetricEJA(n) + sage: x = J.random_element() + sage: y = J.random_element() + sage: z = J.random_element() + sage: (x*y).inner_product(z) == y.inner_product(x*z) + True + """ def __init__(self, n, field=QQ, **kwargs): - S = _real_symmetric_basis(n, field=field) + S = _real_symmetric_basis(n, field) Qs = _multiplication_table_from_matrix_basis(S) fdeja = super(RealSymmetricEJA, self) @@ -1258,9 +1312,20 @@ class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra): sage: ComplexHermitianEJA(2, prefix='z').gens() (z0, z1, z2, z3) + Our inner product satisfies the Jordan axiom:: + + sage: set_random_seed() + sage: n = ZZ.random_element(1,5) + sage: J = ComplexHermitianEJA(n) + sage: x = J.random_element() + sage: y = J.random_element() + sage: z = J.random_element() + sage: (x*y).inner_product(z) == y.inner_product(x*z) + True + """ def __init__(self, n, field=QQ, **kwargs): - S = _complex_hermitian_basis(n) + S = _complex_hermitian_basis(n, field) Qs = _multiplication_table_from_matrix_basis(S) fdeja = super(ComplexHermitianEJA, self) @@ -1324,9 +1389,20 @@ class QuaternionHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra): sage: QuaternionHermitianEJA(2, prefix='a').gens() (a0, a1, a2, a3, a4, a5) + Our inner product satisfies the Jordan axiom:: + + sage: set_random_seed() + sage: n = ZZ.random_element(1,5) + sage: J = QuaternionHermitianEJA(n) + sage: x = J.random_element() + sage: y = J.random_element() + sage: z = J.random_element() + sage: (x*y).inner_product(z) == y.inner_product(x*z) + True + """ def __init__(self, n, field=QQ, **kwargs): - S = _quaternion_hermitian_basis(n) + S = _quaternion_hermitian_basis(n, field) Qs = _multiplication_table_from_matrix_basis(S) fdeja = super(QuaternionHermitianEJA, self) @@ -1386,6 +1462,17 @@ class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra): sage: JordanSpinEJA(2, prefix='B').gens() (B0, B1) + Our inner product satisfies the Jordan axiom:: + + sage: set_random_seed() + sage: n = ZZ.random_element(1,5) + sage: J = JordanSpinEJA(n) + sage: x = J.random_element() + sage: y = J.random_element() + sage: z = J.random_element() + sage: (x*y).inner_product(z) == y.inner_product(x*z) + True + """ def __init__(self, n, field=QQ, **kwargs): V = VectorSpace(field, n)