X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_algebra.py;h=cbe5c08a344a95248e0b7c15e94de80b0c662816;hb=c49406021cb8496ead26ef89865f8afbdc96ea6c;hp=631d695aa75208a65cb457a4ca11280f44823765;hpb=76352ef33974a5ec638b8b1fcab5508f915ea976;p=sage.d.git diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py index 631d695..cbe5c08 100644 --- a/mjo/eja/eja_algebra.py +++ b/mjo/eja/eja_algebra.py @@ -84,25 +84,51 @@ class FiniteDimensionalEJA(CombinatorialFreeModule): INPUT: - - basis -- a tuple of basis elements in "matrix form," which - must be the same form as the arguments to ``jordan_product`` - and ``inner_product``. In reality, "matrix form" can be either - vectors, matrices, or a Cartesian product (ordered tuple) - of vectors or matrices. All of these would ideally be vector - spaces in sage with no special-casing needed; but in reality - we turn vectors into column-matrices and Cartesian products - `(a,b)` into column matrices `(a,b)^{T}` after converting - `a` and `b` themselves. - - - jordan_product -- function of two ``basis`` elements (in - matrix form) that returns their jordan product, also in matrix - form; this will be applied to ``basis`` to compute a - multiplication table for the algebra. - - - inner_product -- function of two ``basis`` elements (in matrix - form) that returns their inner product. This will be applied - to ``basis`` to compute an inner-product table (basically a - matrix) for this algebra. + - ``basis`` -- a tuple; a tuple of basis elements in "matrix + form," which must be the same form as the arguments to + ``jordan_product`` and ``inner_product``. In reality, "matrix + form" can be either vectors, matrices, or a Cartesian product + (ordered tuple) of vectors or matrices. All of these would + ideally be vector spaces in sage with no special-casing + needed; but in reality we turn vectors into column-matrices + and Cartesian products `(a,b)` into column matrices + `(a,b)^{T}` after converting `a` and `b` themselves. + + - ``jordan_product`` -- a function; afunction of two ``basis`` + elements (in matrix form) that returns their jordan product, + also in matrix form; this will be applied to ``basis`` to + compute a multiplication table for the algebra. + + - ``inner_product`` -- a function; a function of two ``basis`` + elements (in matrix form) that returns their inner + product. This will be applied to ``basis`` to compute an + inner-product table (basically a matrix) for this algebra. + + - ``field`` -- a subfield of the reals (default: ``AA``); the scalar + field for the algebra. + + - ``orthonormalize`` -- boolean (default: ``True``); whether or + not to orthonormalize the basis. Doing so is expensive and + generally rules out using the rationals as your ``field``, but + is required for spectral decompositions. + + SETUP:: + + sage: from mjo.eja.eja_algebra import random_eja + + TESTS: + + We should compute that an element subalgebra is associative even + if we circumvent the element method:: + + sage: set_random_seed() + sage: J = random_eja(field=QQ,orthonormalize=False) + sage: x = J.random_element() + sage: A = x.subalgebra_generated_by(orthonormalize=False) + sage: basis = tuple(b.superalgebra_element() for b in A.basis()) + sage: J.subalgebra(basis, orthonormalize=False).is_associative() + True + """ Element = FiniteDimensionalEJAElement @@ -112,7 +138,7 @@ class FiniteDimensionalEJA(CombinatorialFreeModule): inner_product, field=AA, orthonormalize=True, - associative=False, + associative=None, cartesian_product=False, check_field=True, check_axioms=True, @@ -169,7 +195,20 @@ class FiniteDimensionalEJA(CombinatorialFreeModule): category = MagmaticAlgebras(field).FiniteDimensional() - category = category.WithBasis().Unital() + category = category.WithBasis().Unital().Commutative() + + if associative is None: + # We should figure it out. As with check_axioms, we have to do + # this without the help of the _jordan_product_is_associative() + # method because we need to know the category before we + # initialize the algebra. + associative = all( jordan_product(jordan_product(bi,bj),bk) + == + jordan_product(bi,jordan_product(bj,bk)) + for bi in basis + for bj in basis + for bk in basis) + if associative: # Element subalgebras can take advantage of this. category = category.Associative() @@ -413,6 +452,16 @@ class FiniteDimensionalEJA(CombinatorialFreeModule): """ return "Associative" in self.category().axioms() + def _is_commutative(self): + r""" + Whether or not this algebra's multiplication table is commutative. + + This method should of course always return ``True``, unless + this algebra was constructed with ``check_axioms=False`` and + passed an invalid multiplication table. + """ + return all( x*y == y*x for x in self.gens() for y in self.gens() ) + def _is_jordanian(self): r""" Whether or not this algebra's multiplication table respects the @@ -420,7 +469,7 @@ class FiniteDimensionalEJA(CombinatorialFreeModule): We only check one arrangement of `x` and `y`, so for a ``True`` result to be truly true, you should also check - :meth:`is_commutative`. This method should of course always + :meth:`_is_commutative`. This method should of course always return ``True``, unless this algebra was constructed with ``check_axioms=False`` and passed an invalid multiplication table. """ @@ -430,6 +479,92 @@ class FiniteDimensionalEJA(CombinatorialFreeModule): for i in range(self.dimension()) for j in range(self.dimension()) ) + def _jordan_product_is_associative(self): + r""" + Return whether or not this algebra's Jordan product is + associative; that is, whether or not `x*(y*z) = (x*y)*z` + for all `x,y,x`. + + This method should agree with :meth:`is_associative` unless + you lied about the value of the ``associative`` parameter + when you constructed the algebra. + + SETUP:: + + sage: from mjo.eja.eja_algebra import (random_eja, + ....: RealSymmetricEJA, + ....: ComplexHermitianEJA, + ....: QuaternionHermitianEJA) + + EXAMPLES:: + + sage: J = RealSymmetricEJA(4, orthonormalize=False) + sage: J._jordan_product_is_associative() + False + sage: x = sum(J.gens()) + sage: A = x.subalgebra_generated_by() + sage: A._jordan_product_is_associative() + True + + :: + + sage: J = ComplexHermitianEJA(2,field=QQ,orthonormalize=False) + sage: J._jordan_product_is_associative() + False + sage: x = sum(J.gens()) + sage: A = x.subalgebra_generated_by(orthonormalize=False) + sage: A._jordan_product_is_associative() + True + + :: + + sage: J = QuaternionHermitianEJA(2) + sage: J._jordan_product_is_associative() + False + sage: x = sum(J.gens()) + sage: A = x.subalgebra_generated_by() + sage: A._jordan_product_is_associative() + True + + TESTS: + + The values we've presupplied to the constructors agree with + the computation:: + + sage: set_random_seed() + sage: J = random_eja() + sage: J.is_associative() == J._jordan_product_is_associative() + True + + """ + R = self.base_ring() + + # Used to check whether or not something is zero. + epsilon = R.zero() + if not R.is_exact(): + # I don't know of any examples that make this magnitude + # necessary because I don't know how to make an + # associative algebra when the element subalgebra + # construction is unreliable (as it is over RDF; we can't + # find the degree of an element because we can't compute + # the rank of a matrix). But even multiplication of floats + # is non-associative, so *some* epsilon is needed... let's + # just take the one from _inner_product_is_associative? + epsilon = 1e-15 + + for i in range(self.dimension()): + for j in range(self.dimension()): + for k in range(self.dimension()): + x = self.gens()[i] + y = self.gens()[j] + z = self.gens()[k] + diff = (x*y)*z - x*(y*z) + + if diff.norm() > epsilon: + return False + + return True + def _inner_product_is_associative(self): r""" Return whether or not this algebra's inner product `B` is @@ -439,11 +574,14 @@ class FiniteDimensionalEJA(CombinatorialFreeModule): this algebra was constructed with ``check_axioms=False`` and passed an invalid Jordan or inner-product. """ + R = self.base_ring() - # Used to check whether or not something is zero in an inexact - # ring. This number is sufficient to allow the construction of - # QuaternionHermitianEJA(2, field=RDF) with check_axioms=True. - epsilon = 1e-16 + # Used to check whether or not something is zero. + epsilon = R.zero() + if not R.is_exact(): + # This choice is sufficient to allow the construction of + # QuaternionHermitianEJA(2, field=RDF) with check_axioms=True. + epsilon = 1e-15 for i in range(self.dimension()): for j in range(self.dimension()): @@ -453,12 +591,8 @@ class FiniteDimensionalEJA(CombinatorialFreeModule): z = self.gens()[k] diff = (x*y).inner_product(z) - x.inner_product(y*z) - if self.base_ring().is_exact(): - if diff != 0: - return False - else: - if diff.abs() > epsilon: - return False + if diff.abs() > epsilon: + return False return True @@ -472,7 +606,8 @@ class FiniteDimensionalEJA(CombinatorialFreeModule): SETUP:: - sage: from mjo.eja.eja_algebra import (JordanSpinEJA, + sage: from mjo.eja.eja_algebra import (random_eja, + ....: JordanSpinEJA, ....: HadamardEJA, ....: RealSymmetricEJA) @@ -505,18 +640,13 @@ class FiniteDimensionalEJA(CombinatorialFreeModule): TESTS: - Ensure that we can convert any element of the two non-matrix - simple algebras (whose matrix representations are columns) - back and forth faithfully:: + Ensure that we can convert any element back and forth + faithfully between its matrix and algebra representations:: sage: set_random_seed() - sage: J = HadamardEJA.random_instance() - sage: x = J.random_element() - sage: J(x.to_vector().column()) == x - True - sage: J = JordanSpinEJA.random_instance() + sage: J = random_eja() sage: x = J.random_element() - sage: J(x.to_vector().column()) == x + sage: J(x.to_matrix()) == x True We cannot coerce elements between algebras just because their @@ -532,7 +662,6 @@ class FiniteDimensionalEJA(CombinatorialFreeModule): Traceback (most recent call last): ... ValueError: not an element of this algebra - """ msg = "not an element of this algebra" if elt in self.base_ring(): @@ -800,7 +929,7 @@ class FiniteDimensionalEJA(CombinatorialFreeModule): # And to each subsequent row, prepend an entry that belongs to # the left-side "header column." - M += [ [self.gens()[i]] + [ self.product_on_basis(i,j) + M += [ [self.gens()[i]] + [ self.gens()[i]*self.gens()[j] for j in range(n) ] for i in range(n) ] @@ -1471,6 +1600,13 @@ class RationalBasisEJA(FiniteDimensionalEJA): if not all( all(b_i in QQ for b_i in b.list()) for b in basis ): raise TypeError("basis not rational") + super().__init__(basis, + jordan_product, + inner_product, + field=field, + check_field=check_field, + **kwargs) + self._rational_algebra = None if field is not QQ: # There's no point in constructing the extra algebra if this @@ -1484,17 +1620,11 @@ class RationalBasisEJA(FiniteDimensionalEJA): jordan_product, inner_product, field=QQ, + associative=self.is_associative(), orthonormalize=False, check_field=False, check_axioms=False) - super().__init__(basis, - jordan_product, - inner_product, - field=field, - check_field=check_field, - **kwargs) - @cached_method def _charpoly_coefficients(self): r""" @@ -1767,9 +1897,9 @@ class RealSymmetricEJA(ConcreteEJA, RealMatrixEJA): In theory, our "field" can be any subfield of the reals:: - sage: RealSymmetricEJA(2, field=RDF) + sage: RealSymmetricEJA(2, field=RDF, check_axioms=True) Euclidean Jordan algebra of dimension 3 over Real Double Field - sage: RealSymmetricEJA(2, field=RR) + sage: RealSymmetricEJA(2, field=RR, check_axioms=True) Euclidean Jordan algebra of dimension 3 over Real Field with 53 bits of precision @@ -1858,10 +1988,15 @@ class RealSymmetricEJA(ConcreteEJA, RealMatrixEJA): # if the user passes check_axioms=True. if "check_axioms" not in kwargs: kwargs["check_axioms"] = False - super(RealSymmetricEJA, self).__init__(self._denormalized_basis(n), - self.jordan_product, - self.trace_inner_product, - **kwargs) + associative = False + if n <= 1: + associative = True + + super().__init__(self._denormalized_basis(n), + self.jordan_product, + self.trace_inner_product, + associative=associative, + **kwargs) # TODO: this could be factored out somehow, but is left here # because the MatrixEJA is not presently a subclass of the @@ -1951,7 +2086,7 @@ class ComplexMatrixEJA(MatrixEJA): True """ - super(ComplexMatrixEJA,cls).real_embed(M) + super().real_embed(M) n = M.nrows() # We don't need any adjoined elements... @@ -1998,7 +2133,7 @@ class ComplexMatrixEJA(MatrixEJA): True """ - super(ComplexMatrixEJA,cls).real_unembed(M) + super().real_unembed(M) n = ZZ(M.nrows()) d = cls.dimension_over_reals() F = cls.complex_extension(M.base_ring()) @@ -2035,9 +2170,9 @@ class ComplexHermitianEJA(ConcreteEJA, ComplexMatrixEJA): In theory, our "field" can be any subfield of the reals:: - sage: ComplexHermitianEJA(2, field=RDF) + sage: ComplexHermitianEJA(2, field=RDF, check_axioms=True) Euclidean Jordan algebra of dimension 4 over Real Double Field - sage: ComplexHermitianEJA(2, field=RR) + sage: ComplexHermitianEJA(2, field=RR, check_axioms=True) Euclidean Jordan algebra of dimension 4 over Real Field with 53 bits of precision @@ -2146,10 +2281,15 @@ class ComplexHermitianEJA(ConcreteEJA, ComplexMatrixEJA): # if the user passes check_axioms=True. if "check_axioms" not in kwargs: kwargs["check_axioms"] = False - super(ComplexHermitianEJA, self).__init__(self._denormalized_basis(n), - self.jordan_product, - self.trace_inner_product, - **kwargs) + associative = False + if n <= 1: + associative = True + + super().__init__(self._denormalized_basis(n), + self.jordan_product, + self.trace_inner_product, + associative=associative, + **kwargs) # TODO: this could be factored out somehow, but is left here # because the MatrixEJA is not presently a subclass of the # FDEJA class that defines rank() and one(). @@ -2232,7 +2372,7 @@ class QuaternionMatrixEJA(MatrixEJA): True """ - super(QuaternionMatrixEJA,cls).real_embed(M) + super().real_embed(M) quaternions = M.base_ring() n = M.nrows() @@ -2287,7 +2427,7 @@ class QuaternionMatrixEJA(MatrixEJA): True """ - super(QuaternionMatrixEJA,cls).real_unembed(M) + super().real_unembed(M) n = ZZ(M.nrows()) d = cls.dimension_over_reals() @@ -2332,9 +2472,9 @@ class QuaternionHermitianEJA(ConcreteEJA, QuaternionMatrixEJA): In theory, our "field" can be any subfield of the reals:: - sage: QuaternionHermitianEJA(2, field=RDF) + sage: QuaternionHermitianEJA(2, field=RDF, check_axioms=True) Euclidean Jordan algebra of dimension 6 over Real Double Field - sage: QuaternionHermitianEJA(2, field=RR) + sage: QuaternionHermitianEJA(2, field=RR, check_axioms=True) Euclidean Jordan algebra of dimension 6 over Real Field with 53 bits of precision @@ -2452,10 +2592,16 @@ class QuaternionHermitianEJA(ConcreteEJA, QuaternionMatrixEJA): # if the user passes check_axioms=True. if "check_axioms" not in kwargs: kwargs["check_axioms"] = False - super(QuaternionHermitianEJA, self).__init__(self._denormalized_basis(n), - self.jordan_product, - self.trace_inner_product, - **kwargs) + associative = False + if n <= 1: + associative = True + + super().__init__(self._denormalized_basis(n), + self.jordan_product, + self.trace_inner_product, + associative=associative, + **kwargs) + # TODO: this could be factored out somehow, but is left here # because the MatrixEJA is not presently a subclass of the # FDEJA class that defines rank() and one(). @@ -2681,10 +2827,17 @@ class BilinearFormEJA(ConcreteEJA): n = B.nrows() column_basis = tuple( b.column() for b in FreeModule(ZZ, n).basis() ) - super(BilinearFormEJA, self).__init__(column_basis, - jordan_product, - inner_product, - **kwargs) + + # TODO: I haven't actually checked this, but it seems legit. + associative = False + if n <= 2: + associative = True + + super().__init__(column_basis, + jordan_product, + inner_product, + associative=associative, + **kwargs) # The rank of this algebra is two, unless we're in a # one-dimensional ambient space (because the rank is bounded @@ -2789,7 +2942,7 @@ class JordanSpinEJA(BilinearFormEJA): # But also don't pass check_field=False here, because the user # can pass in a field! - super(JordanSpinEJA, self).__init__(B, **kwargs) + super().__init__(B, **kwargs) @staticmethod def _max_random_instance_size(): @@ -2847,10 +3000,12 @@ class TrivialEJA(ConcreteEJA): if "orthonormalize" not in kwargs: kwargs["orthonormalize"] = False if "check_axioms" not in kwargs: kwargs["check_axioms"] = False - super(TrivialEJA, self).__init__(basis, - jordan_product, - inner_product, - **kwargs) + super().__init__(basis, + jordan_product, + inner_product, + associative=True, + **kwargs) + # The rank is zero using my definition, namely the dimension of the # largest subalgebra generated by any element. self.rank.set_cache(0) @@ -3041,6 +3196,65 @@ class CartesianProductEJA(CombinatorialFreeModule_CartesianProduct, self.one.set_cache(self._cartesian_product_of_elements(ones)) self.rank.set_cache(sum(J.rank() for J in algebras)) + def _monomial_to_generator(self, mon): + r""" + Convert a monomial index into a generator index. + + SETUP:: + + sage: from mjo.eja.eja_algebra import random_eja() + + TESTS:: + + sage: J1 = random_eja(field=QQ, orthonormalize=False) + sage: J2 = random_eja(field=QQ, orthonormalize=False) + sage: J = cartesian_product([J1,J2]) + sage: all( J.monomial(m) + ....: == + ....: J.gens()[J._monomial_to_generator(m)] + ....: for m in J.basis().keys() ) + + """ + # The superclass method indexes into a matrix, so we have to + # turn the tuples i and j into integers. This is easy enough + # given that the first coordinate of i and j corresponds to + # the factor, and the second coordinate corresponds to the + # index of the generator within that factor. + factor = mon[0] + idx_in_factor = mon[1] + + offset = sum( f.dimension() + for f in self.cartesian_factors()[:factor] ) + return offset + idx_in_factor + + def product_on_basis(self, i, j): + r""" + Return the product of the monomials indexed by ``i`` and ``j``. + + This overrides the superclass method because here, both ``i`` + and ``j`` will be ordered pairs. + + SETUP:: + + sage: from mjo.eja.eja_algebra import (QuaternionHermitianEJA, + ....: RealSymmetricEJA) + + TESTS:: + + sage: J1 = RealSymmetricEJA(1,field=QQ) + sage: J2 = QuaternionHermitianEJA(1,field=QQ) + sage: J = cartesian_product([J1,J2]) + sage: x = sum(J.gens()) + sage: x == J.one() + True + sage: x*x == x + True + + """ + l = self._monomial_to_generator(i) + m = self._monomial_to_generator(j) + return FiniteDimensionalEJA.product_on_basis(self, l, m) + def matrix_space(self): r""" Return the space that our matrix basis lives in as a Cartesian