from sage.matrix.constructor import matrix from mjo.eja.eja_algebra import FiniteDimensionalEuclideanJordanAlgebra from mjo.eja.eja_element import FiniteDimensionalEuclideanJordanAlgebraElement class FiniteDimensionalEuclideanJordanElementSubalgebraElement(FiniteDimensionalEuclideanJordanAlgebraElement): """ SETUP:: sage: from mjo.eja.eja_algebra import random_eja TESTS:: The natural representation of an element in the subalgebra is the same as its natural representation in the superalgebra:: sage: set_random_seed() sage: A = random_eja().random_element().subalgebra_generated_by() sage: y = A.random_element() sage: actual = y.natural_representation() sage: expected = y.superalgebra_element().natural_representation() sage: actual == expected True """ def __init__(self, A, elt): """ SETUP:: sage: from mjo.eja.eja_algebra import RealSymmetricEJA sage: from mjo.eja.eja_subalgebra import FiniteDimensionalEuclideanJordanElementSubalgebra EXAMPLES:: sage: J = RealSymmetricEJA(3) sage: x = sum( i*J.gens()[i] for i in range(6) ) sage: K = FiniteDimensionalEuclideanJordanElementSubalgebra(x) sage: [ K.element_class(K,x^k) for k in range(J.rank()) ] [f0, f1, f2] :: """ if elt in A.superalgebra(): # Try to convert a parent algebra element into a # subalgebra element... try: coords = A.vector_space().coordinate_vector(elt.to_vector()) elt = A.from_vector(coords).monomial_coefficients() except AttributeError: # Catches a missing method in elt.to_vector() pass s = super(FiniteDimensionalEuclideanJordanElementSubalgebraElement, self) s.__init__(A, elt) def superalgebra_element(self): """ Return the object in our algebra's superalgebra that corresponds to myself. SETUP:: sage: from mjo.eja.eja_algebra import (RealSymmetricEJA, ....: random_eja) EXAMPLES:: sage: J = RealSymmetricEJA(3) sage: x = sum(J.gens()) sage: x e0 + e1 + e2 + e3 + e4 + e5 sage: A = x.subalgebra_generated_by() sage: A.element_class(A,x) f1 sage: A.element_class(A,x).superalgebra_element() e0 + e1 + e2 + e3 + e4 + e5 TESTS: We can convert back and forth faithfully:: sage: set_random_seed() sage: J = random_eja() sage: x = J.random_element() sage: A = x.subalgebra_generated_by() sage: A.element_class(A,x).superalgebra_element() == x True sage: y = A.random_element() sage: A.element_class(A,y.superalgebra_element()) == y True """ return self.parent().superalgebra().linear_combination( zip(self.parent()._superalgebra_basis, self.to_vector()) ) class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclideanJordanAlgebra): """ The subalgebra of an EJA generated by a single element. """ def __init__(self, elt): superalgebra = elt.parent() # First compute the vector subspace spanned by the powers of # the given element. V = superalgebra.vector_space() superalgebra_basis = [superalgebra.one()] basis_vectors = [superalgebra.one().to_vector()] W = V.span_of_basis(basis_vectors) for exponent in range(1, V.dimension()): new_power = elt**exponent basis_vectors.append( new_power.to_vector() ) try: W = V.span_of_basis(basis_vectors) superalgebra_basis.append( new_power ) except ValueError: # Vectors weren't independent; bail and keep the # last subspace that worked. break # Make the basis hashable for UniqueRepresentation. superalgebra_basis = tuple(superalgebra_basis) # Now figure out the entries of the right-multiplication # matrix for the successive basis elements b0, b1,... of # that subspace. field = superalgebra.base_ring() mult_table = [] for b_right in superalgebra_basis: b_right_rows = [] # The first row of the right-multiplication matrix by # b1 is what we get if we apply that matrix to b1. The # second row of the right multiplication matrix by b1 # is what we get when we apply that matrix to b2... # # IMPORTANT: this assumes that all vectors are COLUMN # vectors, unlike our superclass (which uses row vectors). for b_left in superalgebra_basis: # Multiply in the original EJA, but then get the # coordinates from the subalgebra in terms of its # basis. this_row = W.coordinates((b_left*b_right).to_vector()) b_right_rows.append(this_row) b_right_matrix = matrix(field, b_right_rows) mult_table.append(b_right_matrix) for m in mult_table: m.set_immutable() mult_table = tuple(mult_table) # TODO: We'll have to redo this and make it unique again... prefix = 'f' # The rank is the highest possible degree of a minimal # polynomial, and is bounded above by the dimension. We know # in this case that there's an element whose minimal # polynomial has the same degree as the space's dimension # (remember how we constructed the space?), so that must be # its rank too. rank = W.dimension() category = superalgebra.category().Associative() natural_basis = tuple( b.natural_representation() for b in superalgebra_basis ) self._superalgebra = superalgebra self._vector_space = W self._superalgebra_basis = superalgebra_basis fdeja = super(FiniteDimensionalEuclideanJordanElementSubalgebra, self) return fdeja.__init__(field, mult_table, rank, prefix=prefix, category=category, natural_basis=natural_basis) def superalgebra(self): """ Return the superalgebra that this algebra was generated from. """ return self._superalgebra def vector_space(self): """ SETUP:: sage: from mjo.eja.eja_algebra import RealSymmetricEJA sage: from mjo.eja.eja_subalgebra import FiniteDimensionalEuclideanJordanElementSubalgebra EXAMPLES:: sage: J = RealSymmetricEJA(3) sage: x = sum( i*J.gens()[i] for i in range(6) ) sage: K = FiniteDimensionalEuclideanJordanElementSubalgebra(x) sage: K.vector_space() Vector space of degree 6 and dimension 3 over Rational Field User basis matrix: [ 1 0 1 0 0 1] [ 0 1 2 3 4 5] [10 14 21 19 31 50] sage: (x^0).to_vector() (1, 0, 1, 0, 0, 1) sage: (x^1).to_vector() (0, 1, 2, 3, 4, 5) sage: (x^2).to_vector() (10, 14, 21, 19, 31, 50) """ return self._vector_space Element = FiniteDimensionalEuclideanJordanElementSubalgebraElement