from sage.matrix.constructor import matrix from mjo.eja.eja_algebra import FiniteDimensionalEuclideanJordanAlgebra from mjo.eja.eja_element import FiniteDimensionalEuclideanJordanAlgebraElement from mjo.eja.eja_utils import gram_schmidt 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 The left-multiplication-by operator for elements in the subalgebra works like it does in the superalgebra, even if we orthonormalize our basis:: sage: set_random_seed() sage: x = random_eja(AA).random_element() sage: A = x.subalgebra_generated_by(orthonormalize_basis=True) sage: y = A.random_element() sage: y.operator()(A.one()) == y True """ 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.parent()._superalgebra_basis, self.to_vector()) ) class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclideanJordanAlgebra): """ The subalgebra of an EJA generated by a single element. SETUP:: sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA, ....: JordanSpinEJA) TESTS: Ensure that our generator names don't conflict with the superalgebra:: sage: J = JordanSpinEJA(3) sage: J.one().subalgebra_generated_by().gens() (f0,) sage: J = JordanSpinEJA(3, prefix='f') sage: J.one().subalgebra_generated_by().gens() (g0,) sage: J = JordanSpinEJA(3, prefix='b') sage: J.one().subalgebra_generated_by().gens() (c0,) Ensure that we can find subalgebras of subalgebras:: sage: A = ComplexHermitianEJA(3).one().subalgebra_generated_by() sage: B = A.one().subalgebra_generated_by() sage: B.dimension() 1 """ def __init__(self, elt, orthonormalize_basis): self._superalgebra = elt.parent() category = self._superalgebra.category().Associative() V = self._superalgebra.vector_space() field = self._superalgebra.base_ring() # A half-assed attempt to ensure that we don't collide with # the superalgebra's prefix (ignoring the fact that there # could be super-superelgrbas in scope). If possible, we # try to "increment" the parent algebra's prefix, although # this idea goes out the window fast because some prefixen # are off-limits. prefixen = [ 'f', 'g', 'h', 'a', 'b', 'c', 'd' ] try: prefix = prefixen[prefixen.index(self._superalgebra.prefix()) + 1] except ValueError: prefix = prefixen[0] # This list is guaranteed to contain all independent powers, # because it's the maximal set of powers that could possibly # be independent (by a dimension argument). powers = [ elt**k for k in range(V.dimension()) ] if orthonormalize_basis == False: # In this case, we just need to figure out which elements # of the "powers" list are redundant... First compute the # vector subspace spanned by the powers of the given # element. power_vectors = [ p.to_vector() for p in powers ] # Figure out which powers form a linearly-independent set. ind_rows = matrix(field, power_vectors).pivot_rows() # Pick those out of the list of all powers. superalgebra_basis = tuple(map(powers.__getitem__, ind_rows)) # If our superalgebra is a subalgebra of something else, then # these vectors won't have the right coordinates for # V.span_of_basis() unless we use V.from_vector() on them. basis_vectors = map(power_vectors.__getitem__, ind_rows) else: # If we're going to orthonormalize the basis anyway, we # might as well just do Gram-Schmidt on the whole list of # powers. The redundant ones will get zero'd out. superalgebra_basis = gram_schmidt(powers) basis_vectors = [ b.to_vector() for b in superalgebra_basis ] W = V.span_of_basis( V.from_vector(v) for v in basis_vectors ) n = len(superalgebra_basis) mult_table = [[W.zero() for i in range(n)] for j in range(n)] for i in range(n): for j in range(n): product = superalgebra_basis[i]*superalgebra_basis[j] # product.to_vector() might live in a vector subspace # if our parent algebra is already a subalgebra. We # use V.from_vector() to make it "the right size" in # that case. product_vector = V.from_vector(product.to_vector()) mult_table[i][j] = W.coordinate_vector(product_vector) # 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() natural_basis = tuple( b.natural_representation() for b in superalgebra_basis ) 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 _a_regular_element(self): """ Override the superalgebra method to return the one regular element that is sure to exist in this subalgebra, namely the element that generated it. SETUP:: sage: from mjo.eja.eja_algebra import random_eja TESTS:: sage: set_random_seed() sage: J = random_eja().random_element().subalgebra_generated_by() sage: J._a_regular_element().is_regular() True """ if self.dimension() == 0: return self.zero() else: return self.monomial(1) def _element_constructor_(self, elt): """ Construct an element of this subalgebra from the given one. The only valid arguments are elements of the parent algebra that happen to live in this subalgebra. 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,False) sage: [ K(x^k) for k in range(J.rank()) ] [f0, f1, f2] :: """ if elt == 0: # Just as in the superalgebra class, we need to hack # this special case to ensure that random_element() can # coerce a ring zero into the algebra. return self.zero() if elt in self.superalgebra(): coords = self.vector_space().coordinate_vector(elt.to_vector()) return self.from_vector(coords) def one(self): """ Return the multiplicative identity element of this algebra. The superclass method computes the identity element, which is beyond overkill in this case: the superalgebra identity restricted to this algebra is its identity. Note that we can't count on the first basis element being the identity -- it migth have been scaled if we orthonormalized the basis. SETUP:: sage: from mjo.eja.eja_algebra import (RealCartesianProductEJA, ....: random_eja) EXAMPLES:: sage: J = RealCartesianProductEJA(5) sage: J.one() e0 + e1 + e2 + e3 + e4 sage: x = sum(J.gens()) sage: A = x.subalgebra_generated_by() sage: A.one() f0 sage: A.one().superalgebra_element() e0 + e1 + e2 + e3 + e4 TESTS: The identity element acts like the identity over the rationals:: sage: set_random_seed() sage: x = random_eja().random_element() sage: A = x.subalgebra_generated_by() sage: x = A.random_element() sage: A.one()*x == x and x*A.one() == x True The identity element acts like the identity over the algebraic reals with an orthonormal basis:: sage: set_random_seed() sage: x = random_eja(AA).random_element() sage: A = x.subalgebra_generated_by(orthonormalize_basis=True) sage: x = A.random_element() sage: A.one()*x == x and x*A.one() == x True The matrix of the unit element's operator is the identity over the rationals:: sage: set_random_seed() sage: x = random_eja().random_element() sage: A = x.subalgebra_generated_by() sage: actual = A.one().operator().matrix() sage: expected = matrix.identity(A.base_ring(), A.dimension()) sage: actual == expected True The matrix of the unit element's operator is the identity over the algebraic reals with an orthonormal basis:: sage: set_random_seed() sage: x = random_eja(AA).random_element() sage: A = x.subalgebra_generated_by(orthonormalize_basis=True) sage: actual = A.one().operator().matrix() sage: expected = matrix.identity(A.base_ring(), A.dimension()) sage: actual == expected True """ if self.dimension() == 0: return self.zero() else: sa_one = self.superalgebra().one().to_vector() sa_coords = self.vector_space().coordinate_vector(sa_one) return self.from_vector(sa_coords) def natural_basis_space(self): """ Return the natural basis space of this algebra, which is identical to that of its superalgebra. This is correct "by definition," and avoids a mismatch when the subalgebra is trivial (with no natural basis to infer anything from) and the parent is not. """ return self.superalgebra().natural_basis_space() 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 = J.monomial(0) + 2*J.monomial(2) + 5*J.monomial(5) sage: K = FiniteDimensionalEuclideanJordanElementSubalgebra(x,False) sage: K.vector_space() Vector space of degree 6 and dimension 3 over... User basis matrix: [ 1 0 1 0 0 1] [ 1 0 2 0 0 5] [ 1 0 4 0 0 25] sage: (x^0).to_vector() (1, 0, 1, 0, 0, 1) sage: (x^1).to_vector() (1, 0, 2, 0, 0, 5) sage: (x^2).to_vector() (1, 0, 4, 0, 0, 25) """ return self._vector_space Element = FiniteDimensionalEuclideanJordanElementSubalgebraElement