from sage.matrix.constructor import matrix from sage.structure.category_object import normalize_names from mjo.eja.eja_algebra import FiniteDimensionalEuclideanJordanAlgebra from mjo.eja.eja_element import FiniteDimensionalEuclideanJordanAlgebraElement class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclideanJordanAlgebra): """ The subalgebra of an EJA generated by a single element. SETUP:: sage: from mjo.eja.eja_algebra import FiniteDimensionalEuclideanJordanAlgebra TESTS: Ensure that non-clashing names are chosen:: sage: m1 = matrix.identity(QQ,2) sage: m2 = matrix(QQ, [[0,1], ....: [1,0]]) sage: J = FiniteDimensionalEuclideanJordanAlgebra(QQ, ....: [m1,m2], ....: 2, ....: names='f') sage: J.variable_names() ('f0', 'f1') sage: A = sum(J.gens()).subalgebra_generated_by() sage: A.variable_names() ('g0', 'g1') """ @staticmethod def __classcall_private__(cls, 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().vector()] W = V.span_of_basis(basis_vectors) for exponent in range(1, V.dimension()): new_power = elt**exponent basis_vectors.append( new_power.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. F = 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).vector()) b_right_rows.append(this_row) b_right_matrix = matrix(F, b_right_rows) mult_table.append(b_right_matrix) for m in mult_table: m.set_immutable() mult_table = tuple(mult_table) # 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() # EJAs are power-associative, and this algebra is nothin but # powers. assume_associative=True # Figure out a non-conflicting set of names to use. valid_names = ['f','g','h','a','b','c','d'] name_idx = 0 names = normalize_names(W.dimension(), valid_names[0]) # This loops so long as the list of collisions is nonempty. # Just crash if we run out of names without finding a set that # don't conflict with the parent algebra. while [y for y in names if y in superalgebra.variable_names()]: name_idx += 1 names = normalize_names(W.dimension(), valid_names[name_idx]) cat = superalgebra.category().Associative() natural_basis = tuple( b.natural_representation() for b in superalgebra_basis ) fdeja = super(FiniteDimensionalEuclideanJordanElementSubalgebra, cls) return fdeja.__classcall__(cls, F, mult_table, rank, superalgebra_basis, W, assume_associative=assume_associative, names=names, category=cat, natural_basis=natural_basis) def __init__(self, field, mult_table, rank, superalgebra_basis, vector_space, assume_associative=True, names='f', category=None, natural_basis=None): self._superalgebra = superalgebra_basis[0].parent() self._vector_space = vector_space self._superalgebra_basis = superalgebra_basis fdeja = super(FiniteDimensionalEuclideanJordanElementSubalgebra, self) fdeja.__init__(field, mult_table, rank, assume_associative=assume_associative, names=names, 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 0 1 0 1] [ 0 1 2 3 4 5] [ 5 11 14 26 34 45] sage: (x^0).vector() (1, 0, 0, 1, 0, 1) sage: (x^1).vector() (0, 1, 2, 3, 4, 5) sage: (x^2).vector() (5, 11, 14, 26, 34, 45) """ return self._vector_space class Element(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=None): """ 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(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().coordinates(elt.vector()) elt = A(coords) except AttributeError: # Catches a missing method in elt.vector() pass FiniteDimensionalEuclideanJordanAlgebraElement.__init__(self, 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(x) f1 sage: 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(x).superalgebra_element() == x True sage: y = A.random_element() sage: A(y.superalgebra_element()) == y True """ return self.parent().superalgebra().linear_combination( zip(self.vector(), self.parent()._superalgebra_basis) )