X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_algebra.py;h=1fc3618005eb0ac8be12e0e3e09849ab6f983819;hb=ad25c5b8995a1cacefbf4d677316b9e7069521ff;hp=9aa40eeacdacbb54c792158b1462d2f36d3dc4d6;hpb=d3e40bda3f50a2101103192b91b2ab2a911c0311;p=sage.d.git diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py index 9aa40ee..1fc3618 100644 --- a/mjo/eja/eja_algebra.py +++ b/mjo/eja/eja_algebra.py @@ -57,11 +57,15 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): prefix='e', category=None, natural_basis=None, - check=True): + check_field=True, + check_axioms=True): """ SETUP:: - sage: from mjo.eja.eja_algebra import (JordanSpinEJA, random_eja) + sage: from mjo.eja.eja_algebra import ( + ....: FiniteDimensionalEuclideanJordanAlgebra, + ....: JordanSpinEJA, + ....: random_eja) EXAMPLES: @@ -75,20 +79,33 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): TESTS: - The ``field`` we're given must be real:: + The ``field`` we're given must be real with ``check_field=True``:: sage: JordanSpinEJA(2,QQbar) Traceback (most recent call last): ... - ValueError: field is not real + ValueError: scalar field is not real + + The multiplication table must be square with ``check_axioms=True``:: + + sage: FiniteDimensionalEuclideanJordanAlgebra(QQ,((),())) + Traceback (most recent call last): + ... + ValueError: multiplication table is not square """ - if check: + if check_field: if not field.is_subring(RR): # Note: this does return true for the real algebraic - # field, and any quadratic field where we've specified - # a real embedding. - raise ValueError('field is not real') + # field, the rationals, and any quadratic field where + # we've specified a real embedding. + raise ValueError("scalar field is not real") + + # The multiplication table had better be square + n = len(mult_table) + if check_axioms: + if not all( len(l) == n for l in mult_table ): + raise ValueError("multiplication table is not square") self._natural_basis = natural_basis @@ -98,7 +115,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): fda = super(FiniteDimensionalEuclideanJordanAlgebra, self) fda.__init__(field, - range(len(mult_table)), + range(n), prefix=prefix, category=category) self.print_options(bracket='') @@ -114,6 +131,13 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): for ls in mult_table ] + if check_axioms: + if not self._is_commutative(): + raise ValueError("algebra is not commutative") + if not self._is_jordanian(): + raise ValueError("Jordan identity does not hold") + if not self._inner_product_is_associative(): + raise ValueError("inner product is not associative") def _element_constructor_(self, elt): """ @@ -235,6 +259,67 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): def product_on_basis(self, i, j): return self._multiplication_table[i][j] + 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( self.product_on_basis(i,j) == self.product_on_basis(i,j) + for i in range(self.dimension()) + for j in range(self.dimension()) ) + + def _is_jordanian(self): + r""" + Whether or not this algebra's multiplication table respects the + Jordan identity `(x^{2})(xy) = x(x^{2}y)`. + + 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 + return ``True``, unless this algebra was constructed with + ``check_axioms=False`` and passed an invalid multiplication table. + """ + return all( (self.monomial(i)**2)*(self.monomial(i)*self.monomial(j)) + == + (self.monomial(i))*((self.monomial(i)**2)*self.monomial(j)) + for i in range(self.dimension()) + for j in range(self.dimension()) ) + + def _inner_product_is_associative(self): + r""" + Return whether or not this algebra's inner product `B` is + associative; that is, whether or not `B(xy,z) = B(x,yz)`. + + This method should of course always return ``True``, unless + this algebra was constructed with ``check_axioms=False`` and + passed an invalid multiplication table. + """ + + # Used to check whether or not something is zero in an inexact + # ring. This number is sufficient to allow the construction of + # QuaternionHermitianEJA(2, RDF) with check_axioms=True. + epsilon = 1e-16 + + for i in range(self.dimension()): + for j in range(self.dimension()): + for k in range(self.dimension()): + x = self.monomial(i) + y = self.monomial(j) + z = self.monomial(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 + + return True + @cached_method def characteristic_polynomial_of(self): """ @@ -581,6 +666,25 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): Vector space of degree 6 and dimension 2... sage: J1 Euclidean Jordan algebra of dimension 3... + sage: J0.one().natural_representation() + [0 0 0] + [0 0 0] + [0 0 1] + sage: orig_df = AA.options.display_format + sage: AA.options.display_format = 'radical' + sage: J.from_vector(J5.basis()[0]).natural_representation() + [ 0 0 1/2*sqrt(2)] + [ 0 0 0] + [1/2*sqrt(2) 0 0] + sage: J.from_vector(J5.basis()[1]).natural_representation() + [ 0 0 0] + [ 0 0 1/2*sqrt(2)] + [ 0 1/2*sqrt(2) 0] + sage: AA.options.display_format = orig_df + sage: J1.one().natural_representation() + [1 0 0] + [0 1 0] + [0 0 0] TESTS: @@ -595,9 +699,10 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): sage: J1.superalgebra() == J and J1.dimension() == J.dimension() True - The identity elements in the two subalgebras are the - projections onto their respective subspaces of the - superalgebra's identity element:: + The decomposition is into eigenspaces, and its components are + therefore necessarily orthogonal. Moreover, the identity + elements in the two subalgebras are the projections onto their + respective subspaces of the superalgebra's identity element:: sage: set_random_seed() sage: J = random_eja() @@ -607,6 +712,16 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): ....: x = J.random_element() sage: c = x.subalgebra_idempotent() sage: J0,J5,J1 = J.peirce_decomposition(c) + sage: ipsum = 0 + sage: for (w,y,z) in zip(J0.basis(), J5.basis(), J1.basis()): + ....: w = w.superalgebra_element() + ....: y = J.from_vector(y) + ....: z = z.superalgebra_element() + ....: ipsum += w.inner_product(y).abs() + ....: ipsum += w.inner_product(z).abs() + ....: ipsum += y.inner_product(z).abs() + sage: ipsum + 0 sage: J1(c) == J1.one() True sage: J0(J.one() - c) == J0.one() @@ -631,7 +746,9 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): J5 = eigspace else: gens = tuple( self.from_vector(b) for b in eigspace.basis() ) - subalg = FiniteDimensionalEuclideanJordanSubalgebra(self, gens) + subalg = FiniteDimensionalEuclideanJordanSubalgebra(self, + gens, + check_axioms=False) if eigval == 0: J0 = subalg elif eigval == 1: @@ -642,10 +759,61 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): return (J0, J5, J1) - def random_elements(self, count): + def random_element(self, thorough=False): + r""" + Return a random element of this algebra. + + Our algebra superclass method only returns a linear + combination of at most two basis elements. We instead + want the vector space "random element" method that + returns a more diverse selection. + + INPUT: + + - ``thorough`` -- (boolean; default False) whether or not we + should generate irrational coefficients for the random + element when our base ring is irrational; this slows the + algebra operations to a crawl, but any truly random method + should include them + + """ + # For a general base ring... maybe we can trust this to do the + # right thing? Unlikely, but. + V = self.vector_space() + v = V.random_element() + + if self.base_ring() is AA: + # The "random element" method of the algebraic reals is + # stupid at the moment, and only returns integers between + # -2 and 2, inclusive: + # + # https://trac.sagemath.org/ticket/30875 + # + # Instead, we implement our own "random vector" method, + # and then coerce that into the algebra. We use the vector + # space degree here instead of the dimension because a + # subalgebra could (for example) be spanned by only two + # vectors, each with five coordinates. We need to + # generate all five coordinates. + if thorough: + v *= QQbar.random_element().real() + else: + v *= QQ.random_element() + + return self.from_vector(V.coordinate_vector(v)) + + def random_elements(self, count, thorough=False): """ Return ``count`` random elements as a tuple. + INPUT: + + - ``thorough`` -- (boolean; default False) whether or not we + should generate irrational coefficients for the random + elements when our base ring is irrational; this slows the + algebra operations to a crawl, but any truly random method + should include them + SETUP:: sage: from mjo.eja.eja_algebra import JordanSpinEJA @@ -660,7 +828,8 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): True """ - return tuple( self.random_element() for idx in range(count) ) + return tuple( self.random_element(thorough) + for idx in range(count) ) @classmethod def random_instance(cls, field=AA, **kwargs): @@ -838,79 +1007,6 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): Element = FiniteDimensionalEuclideanJordanAlgebraElement -class HadamardEJA(FiniteDimensionalEuclideanJordanAlgebra): - """ - Return the Euclidean Jordan Algebra corresponding to the set - `R^n` under the Hadamard product. - - Note: this is nothing more than the Cartesian product of ``n`` - copies of the spin algebra. Once Cartesian product algebras - are implemented, this can go. - - SETUP:: - - sage: from mjo.eja.eja_algebra import HadamardEJA - - EXAMPLES: - - This multiplication table can be verified by hand:: - - sage: J = HadamardEJA(3) - sage: e0,e1,e2 = J.gens() - sage: e0*e0 - e0 - sage: e0*e1 - 0 - sage: e0*e2 - 0 - sage: e1*e1 - e1 - sage: e1*e2 - 0 - sage: e2*e2 - e2 - - TESTS: - - We can change the generator prefix:: - - sage: HadamardEJA(3, prefix='r').gens() - (r0, r1, r2) - - """ - def __init__(self, n, field=AA, **kwargs): - V = VectorSpace(field, n) - mult_table = [ [ V.gen(i)*(i == j) for j in range(n) ] - for i in range(n) ] - - fdeja = super(HadamardEJA, self) - fdeja.__init__(field, mult_table, **kwargs) - self.rank.set_cache(n) - - def inner_product(self, x, y): - """ - Faster to reimplement than to use natural representations. - - SETUP:: - - sage: from mjo.eja.eja_algebra import HadamardEJA - - TESTS: - - Ensure that this is the usual inner product for the algebras - over `R^n`:: - - sage: set_random_seed() - sage: J = HadamardEJA.random_instance() - sage: x,y = J.random_elements(2) - sage: X = x.natural_representation() - sage: Y = y.natural_representation() - sage: x.inner_product(y) == J.natural_inner_product(X,Y) - True - - """ - return x.to_vector().inner_product(y.to_vector()) - def random_eja(field=AA): """ @@ -937,6 +1033,44 @@ def random_eja(field=AA): +class RationalBasisEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra): + r""" + Algebras whose basis consists of vectors with rational + entries. Equivalently, algebras whose multiplication tables + contain only rational coefficients. + + When an EJA has a basis that can be made rational, we can speed up + the computation of its characteristic polynomial by doing it over + ``QQ``. All of the named EJA constructors that we provide fall + into this category. + """ + @cached_method + def _charpoly_coefficients(self): + r""" + Override the parent method with something that tries to compute + over a faster (non-extension) field. + """ + if self.base_ring() is QQ: + # There's no need to construct *another* algebra over the + # rationals if this one is already over the rationals. + superclass = super(RationalBasisEuclideanJordanAlgebra, self) + return superclass._charpoly_coefficients() + + mult_table = tuple( + map(lambda x: x.to_vector(), ls) + for ls in self._multiplication_table + ) + + # Do the computation over the rationals. The answer will be + # the same, because our basis coordinates are (essentially) + # rational. + J = FiniteDimensionalEuclideanJordanAlgebra(QQ, + mult_table, + check_field=False, + check_axioms=False) + return J._charpoly_coefficients() + + class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra): @staticmethod def _max_test_case_size(): @@ -973,9 +1107,10 @@ class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra): Qs = self.multiplication_table_from_matrix_basis(basis) - fdeja = super(MatrixEuclideanJordanAlgebra, self) - fdeja.__init__(field, Qs, natural_basis=basis, **kwargs) - return + super(MatrixEuclideanJordanAlgebra, self).__init__(field, + Qs, + natural_basis=basis, + **kwargs) @cached_method @@ -994,10 +1129,14 @@ class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra): # Do this over the rationals and convert back at the end. # Only works because we know the entries of the basis are - # integers. + # integers. The argument ``check_axioms=False`` is required + # because the trace inner-product method for this + # class is a stub and can't actually be checked. J = MatrixEuclideanJordanAlgebra(QQ, basis, - normalize_basis=False) + normalize_basis=False, + check_field=False, + check_axioms=False) a = J._charpoly_coefficients() # Unfortunately, changing the basis does change the @@ -1084,16 +1223,11 @@ class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra): Yu = cls.real_unembed(Y) tr = (Xu*Yu).trace() - if tr in RLF: - # It's real already. - return tr - - # Otherwise, try the thing that works for complex numbers; and - # if that doesn't work, the thing that works for quaternions. try: - return tr.vector()[0] # real part, imag part is index 1 + # Works in QQ, AA, RDF, et cetera. + return tr.real() except AttributeError: - # A quaternions doesn't have a vector() method, but does + # A quaternion doesn't have a real() method, but does # have coefficient_tuple() method that returns the # coefficients of 1, i, j, and k -- in that order. return tr.coefficient_tuple()[0] @@ -1240,7 +1374,10 @@ class RealSymmetricEJA(RealMatrixEuclideanJordanAlgebra): def __init__(self, n, field=AA, **kwargs): basis = self._denormalized_basis(n, field) - super(RealSymmetricEJA, self).__init__(field, basis, **kwargs) + super(RealSymmetricEJA, self).__init__(field, + basis, + check_axioms=False, + **kwargs) self.rank.set_cache(n) @@ -1536,7 +1673,10 @@ class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra): def __init__(self, n, field=AA, **kwargs): basis = self._denormalized_basis(n,field) - super(ComplexHermitianEJA,self).__init__(field, basis, **kwargs) + super(ComplexHermitianEJA,self).__init__(field, + basis, + check_axioms=False, + **kwargs) self.rank.set_cache(n) @@ -1837,11 +1977,90 @@ class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra): def __init__(self, n, field=AA, **kwargs): basis = self._denormalized_basis(n,field) - super(QuaternionHermitianEJA,self).__init__(field, basis, **kwargs) + super(QuaternionHermitianEJA,self).__init__(field, + basis, + check_axioms=False, + **kwargs) self.rank.set_cache(n) -class BilinearFormEJA(FiniteDimensionalEuclideanJordanAlgebra): +class HadamardEJA(RationalBasisEuclideanJordanAlgebra): + """ + Return the Euclidean Jordan Algebra corresponding to the set + `R^n` under the Hadamard product. + + Note: this is nothing more than the Cartesian product of ``n`` + copies of the spin algebra. Once Cartesian product algebras + are implemented, this can go. + + SETUP:: + + sage: from mjo.eja.eja_algebra import HadamardEJA + + EXAMPLES: + + This multiplication table can be verified by hand:: + + sage: J = HadamardEJA(3) + sage: e0,e1,e2 = J.gens() + sage: e0*e0 + e0 + sage: e0*e1 + 0 + sage: e0*e2 + 0 + sage: e1*e1 + e1 + sage: e1*e2 + 0 + sage: e2*e2 + e2 + + TESTS: + + We can change the generator prefix:: + + sage: HadamardEJA(3, prefix='r').gens() + (r0, r1, r2) + + """ + def __init__(self, n, field=AA, **kwargs): + V = VectorSpace(field, n) + mult_table = [ [ V.gen(i)*(i == j) for j in range(n) ] + for i in range(n) ] + + super(HadamardEJA, self).__init__(field, + mult_table, + check_axioms=False, + **kwargs) + self.rank.set_cache(n) + + def inner_product(self, x, y): + """ + Faster to reimplement than to use natural representations. + + SETUP:: + + sage: from mjo.eja.eja_algebra import HadamardEJA + + TESTS: + + Ensure that this is the usual inner product for the algebras + over `R^n`:: + + sage: set_random_seed() + sage: J = HadamardEJA.random_instance() + sage: x,y = J.random_elements(2) + sage: X = x.natural_representation() + sage: Y = y.natural_representation() + sage: x.inner_product(y) == J.natural_inner_product(X,Y) + True + + """ + return x.to_vector().inner_product(y.to_vector()) + + +class BilinearFormEJA(RationalBasisEuclideanJordanAlgebra): r""" The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)`` with the half-trace inner product and jordan product ``x*y = @@ -1920,8 +2139,10 @@ class BilinearFormEJA(FiniteDimensionalEuclideanJordanAlgebra): # The rank of this algebra is two, unless we're in a # one-dimensional ambient space (because the rank is bounded # by the ambient dimension). - fdeja = super(BilinearFormEJA, self) - fdeja.__init__(field, mult_table, **kwargs) + super(BilinearFormEJA, self).__init__(field, + mult_table, + check_axioms=False, + **kwargs) self.rank.set_cache(min(n,2)) def inner_product(self, x, y): @@ -2013,7 +2234,7 @@ class JordanSpinEJA(BilinearFormEJA): def __init__(self, n, field=AA, **kwargs): # This is a special case of the BilinearFormEJA with the identity # matrix as its bilinear form. - return super(JordanSpinEJA, self).__init__(n, field, **kwargs) + super(JordanSpinEJA, self).__init__(n, field, **kwargs) class TrivialEJA(FiniteDimensionalEuclideanJordanAlgebra): @@ -2047,8 +2268,59 @@ class TrivialEJA(FiniteDimensionalEuclideanJordanAlgebra): """ def __init__(self, field=AA, **kwargs): mult_table = [] - fdeja = super(TrivialEJA, self) + super(TrivialEJA, self).__init__(field, + mult_table, + check_axioms=False, + **kwargs) # The rank is zero using my definition, namely the dimension of the # largest subalgebra generated by any element. - fdeja.__init__(field, mult_table, **kwargs) self.rank.set_cache(0) + + +class DirectSumEJA(FiniteDimensionalEuclideanJordanAlgebra): + r""" + The external (orthogonal) direct sum of two other Euclidean Jordan + algebras. Essentially the Cartesian product of its two factors. + Every Euclidean Jordan algebra decomposes into an orthogonal + direct sum of simple Euclidean Jordan algebras, so no generality + is lost by providing only this construction. + + SETUP:: + + sage: from mjo.eja.eja_algebra import (HadamardEJA, + ....: RealSymmetricEJA, + ....: DirectSumEJA) + + EXAMPLES:: + + sage: J1 = HadamardEJA(2) + sage: J2 = RealSymmetricEJA(3) + sage: J = DirectSumEJA(J1,J2) + sage: J.dimension() + 8 + sage: J.rank() + 5 + + """ + def __init__(self, J1, J2, field=AA, **kwargs): + n1 = J1.dimension() + n2 = J2.dimension() + n = n1+n2 + V = VectorSpace(field, n) + mult_table = [ [ V.zero() for j in range(n) ] + for i in range(n) ] + for i in range(n1): + for j in range(n1): + p = (J1.monomial(i)*J1.monomial(j)).to_vector() + mult_table[i][j] = V(p.list() + [field.zero()]*n2) + + for i in range(n2): + for j in range(n2): + p = (J2.monomial(i)*J2.monomial(j)).to_vector() + mult_table[n1+i][n1+j] = V([field.zero()]*n1 + p.list()) + + super(DirectSumEJA, self).__init__(field, + mult_table, + check_axioms=False, + **kwargs) + self.rank.set_cache(J1.rank() + J2.rank())