X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_algebra.py;h=0da3eef9aa6e7835b6a6291d008bed4e31571872;hb=2b95649f57f150fed77ef2e62076eb97f54fa6da;hp=91088a2eb799ce616bfffc5ee3e3980afe17f4ac;hpb=07f5168dc8e9434171ab89d7089330c561da4380;p=sage.d.git diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py index 91088a2..0da3eef 100644 --- a/mjo/eja/eja_algebra.py +++ b/mjo/eja/eja_algebra.py @@ -61,7 +61,10 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): """ SETUP:: - sage: from mjo.eja.eja_algebra import (JordanSpinEJA, random_eja) + sage: from mjo.eja.eja_algebra import ( + ....: FiniteDimensionalEuclideanJordanAlgebra, + ....: JordanSpinEJA, + ....: random_eja) EXAMPLES: @@ -75,13 +78,20 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): TESTS: - The ``field`` we're given must be real:: + The ``field`` we're given must be real with ``check=True``:: sage: JordanSpinEJA(2,QQbar) Traceback (most recent call last): ... ValueError: field is not real + The multiplication table must be square with ``check=True``:: + + sage: FiniteDimensionalEuclideanJordanAlgebra(QQ,((),())) + Traceback (most recent call last): + ... + ValueError: multiplication table is not square + """ if check: if not field.is_subring(RR): @@ -96,9 +106,15 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): category = MagmaticAlgebras(field).FiniteDimensional() category = category.WithBasis().Unital() + # The multiplication table had better be square + n = len(mult_table) + if check: + if not all( len(l) == n for l in mult_table ): + raise ValueError("multiplication table is not square") + fda = super(FiniteDimensionalEuclideanJordanAlgebra, self) fda.__init__(field, - range(len(mult_table)), + range(n), prefix=prefix, category=category) self.print_options(bracket='') @@ -114,6 +130,13 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): for ls in mult_table ] + if check: + 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 +258,67 @@ 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=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=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=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=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): """ @@ -698,12 +782,16 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): 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. 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. + # -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: @@ -1072,10 +1160,13 @@ 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=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=False) a = J._charpoly_coefficients() # Unfortunately, changing the basis does change the @@ -2130,3 +2221,50 @@ class TrivialEJA(FiniteDimensionalEuclideanJordanAlgebra): # 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()) + + fdeja = super(DirectSumEJA, self) + fdeja.__init__(field, mult_table, **kwargs) + self.rank.set_cache(J1.rank() + J2.rank())