X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_algebra.py;h=389efbf77219acff73db032e848e34d8da1670e9;hb=9632a965029feadbb70d18574f8704812ed71321;hp=374af5498160180ff449d84c0d59d9f2f1db5f42;hpb=6e0da798fd89daa493d06ac0af45fbd989d4ac28;p=sage.d.git diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py index 374af54..389efbf 100644 --- a/mjo/eja/eja_algebra.py +++ b/mjo/eja/eja_algebra.py @@ -773,6 +773,108 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): return (J0, J5, J1) + def a_jordan_frame(self): + r""" + Generate a Jordan frame for this algebra. + + This implementation is based on the so-called "central + orthogonal idempotents" implemented for (semisimple) centers + of SageMath ``FiniteDimensionalAlgebrasWithBasis``. Since all + Euclidean Jordan algebas are commutative (and thus equal to + their own centers) and semisimple, the method should work more + or less as implemented, if it ever worked in the first place. + (I don't know the justification for the original implementation. + yet). + + How it works: we loop through the algebras generators, looking + for their eigenspaces. If there's more than one eigenspace, + and if they result in more than one subalgebra, then we split + those subalgebras recursively until we get to subalgebras of + dimension one (whose idempotent is the unit element). Why does + some generator have to produce at least two subalgebras? I + dunno. But it seems to work. + + Beware that Koecher defines the "center" of a Jordan algebra to + be something else, because the usual definition is stupid in a + (necessarily commutative) Jordan algebra. + + SETUP:: + + sage: from mjo.eja.eja_algebra import (random_eja, + ....: JordanSpinEJA, + ....: TrivialEJA) + + EXAMPLES: + + A Jordan frame for the trivial algebra has to be empty + (zero-length) since its rank is zero. More to the point, there + are no non-zero idempotents in the trivial EJA. This does not + cause any problems so long as we adopt the convention that the + empty sum is zero, since then the sole element of the trivial + EJA has an (empty) spectral decomposition:: + + sage: J = TrivialEJA() + sage: J.a_jordan_frame() + () + + A one-dimensional algebra has rank one (equal to its dimension), + and only one primitive idempotent, namely the algebra's unit + element:: + + sage: J = JordanSpinEJA(1) + sage: J.a_jordan_frame() + (e0,) + + TESTS:: + + sage: J = random_eja() + sage: c = J.a_jordan_frame() + sage: all( x^2 == x for x in c ) + True + sage: r = len(c) + sage: all( c[i]*c[j] == c[i]*(i==j) for i in range(r) + ....: for j in range(r) ) + True + + """ + if self.dimension() == 0: + return () + if self.dimension() == 1: + return (self.one(),) + + for g in self.gens(): + eigenpairs = g.operator().matrix().right_eigenspaces() + if len(eigenpairs) >= 2: + subalgebras = [] + for eigval, eigspace in eigenpairs: + # Make sub-EJAs from the matrix eigenspaces... + sb = tuple( self.from_vector(b) for b in eigspace.basis() ) + try: + # This will fail if e.g. the eigenspace basis + # contains two elements and their product + # isn't a linear combination of the two of + # them (i.e. the generated EJA isn't actually + # two dimensional). + s = FiniteDimensionalEuclideanJordanSubalgebra(self, sb) + subalgebras.append(s) + except ArithmeticError as e: + if str(e) == "vector is not in free module": + # Ignore only the "not a sub-EJA" error + pass + + if len(subalgebras) >= 2: + # apply this method recursively. + return tuple( c.superalgebra_element() + for subalgebra in subalgebras + for c in subalgebra.a_jordan_frame() ) + + # If we got here, the algebra didn't decompose, at least not when we looked at + # the eigenspaces corresponding only to basis elements of the algebra. The + # implementation I stole says that this should work because of Schur's Lemma, + # so I personally blame Schur's Lemma if it does not. + raise Exception("Schur's Lemma didn't work!") + + def random_elements(self, count): """ Return ``count`` random elements as a tuple. @@ -2017,7 +2119,7 @@ class BilinearFormEJA(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA): return x[0]*y[0] + (self._B*xbar).inner_product(ybar) -class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA): +class JordanSpinEJA(BilinearFormEJA): """ The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)`` with the usual inner product and jordan product ``x*y = @@ -2054,42 +2156,9 @@ class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA): sage: JordanSpinEJA(2, prefix='B').gens() (B0, B1) - """ - def __init__(self, n, field=QQ, **kwargs): - V = VectorSpace(field, n) - mult_table = [[V.zero() for j in range(n)] for i in range(n)] - for i in range(n): - for j in range(n): - x = V.gen(i) - y = V.gen(j) - x0 = x[0] - xbar = x[1:] - y0 = y[0] - ybar = y[1:] - # z = x*y - z0 = x.inner_product(y) - zbar = y0*xbar + x0*ybar - z = V([z0] + zbar.list()) - mult_table[i][j] = z - - # The rank of the spin algebra is two, unless we're in a - # one-dimensional ambient space (because the rank is bounded by - # the ambient dimension). - fdeja = super(JordanSpinEJA, self) - return fdeja.__init__(field, mult_table, rank=min(n,2), **kwargs) - - def inner_product(self, x, y): - """ - Faster to reimplement than to use natural representations. - - SETUP:: - - sage: from mjo.eja.eja_algebra import JordanSpinEJA - - TESTS: + TESTS: - Ensure that this is the usual inner product for the algebras - over `R^n`:: + Ensure that we have the usual inner product on `R^n`:: sage: set_random_seed() sage: J = JordanSpinEJA.random_instance() @@ -2099,8 +2168,11 @@ class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA): sage: x.inner_product(y) == J.natural_inner_product(X,Y) True - """ - return x.to_vector().inner_product(y.to_vector()) + """ + def __init__(self, n, field=QQ, **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) class TrivialEJA(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA):