X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_algebra.py;h=01deb1b65140675ffbc81c9c673c21e6d4a2e77f;hb=8e4c84f17025f6c3f2d049902a6a01f954d484ee;hp=9198a8b124e041e4edcc4351271db01345accaa0;hpb=9dd1e4c84fa17c8fe9d758a4fec9e090965c5cb9;p=sage.d.git diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py index 9198a8b..01deb1b 100644 --- a/mjo/eja/eja_algebra.py +++ b/mjo/eja/eja_algebra.py @@ -17,7 +17,7 @@ from sage.misc.lazy_import import lazy_import from sage.misc.prandom import choice from sage.misc.table import table from sage.modules.free_module import FreeModule, VectorSpace -from sage.rings.all import (ZZ, QQ, RR, RLF, CLF, +from sage.rings.all import (ZZ, QQ, AA, QQbar, RR, RLF, CLF, PolynomialRing, QuadraticField) from mjo.eja.eja_element import FiniteDimensionalEuclideanJordanAlgebraElement @@ -54,7 +54,6 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): def __init__(self, field, mult_table, - rank, prefix='e', category=None, natural_basis=None, @@ -91,7 +90,6 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): # a real embedding. raise ValueError('field is not real') - self._rank = rank self._natural_basis = natural_basis if category is None: @@ -194,6 +192,24 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): coords = W.coordinate_vector(_mat2vec(elt)) return self.from_vector(coords) + @staticmethod + def _max_test_case_size(): + """ + Return an integer "size" that is an upper bound on the size of + this algebra when it is used in a random test + case. Unfortunately, the term "size" is quite vague -- when + dealing with `R^n` under either the Hadamard or Jordan spin + product, the "size" refers to the dimension `n`. When dealing + with a matrix algebra (real symmetric or complex/quaternion + Hermitian), it refers to the size of the matrix, which is + far less than the dimension of the underlying vector space. + + We default to five in this class, which is safe in `R^n`. The + matrix algebra subclasses (or any class where the "size" is + interpreted to be far less than the dimension) should override + with a smaller number. + """ + return 5 def _repr_(self): """ @@ -207,8 +223,8 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): Ensure that it says what we think it says:: - sage: JordanSpinEJA(2, field=QQ) - Euclidean Jordan algebra of dimension 2 over Rational Field + sage: JordanSpinEJA(2, field=AA) + Euclidean Jordan algebra of dimension 2 over Algebraic Real Field sage: JordanSpinEJA(3, field=RDF) Euclidean Jordan algebra of dimension 3 over Real Double Field @@ -551,8 +567,8 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): Finite family {0: e0, 1: e1, 2: e2} sage: J.natural_basis() ( - [1 0] [ 0 1/2*sqrt2] [0 0] - [0 0], [1/2*sqrt2 0], [0 1] + [1 0] [ 0 0.7071067811865475?] [0 0] + [0 0], [0.7071067811865475? 0], [0 1] ) :: @@ -757,7 +773,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): J5 = VectorSpace(self.base_ring(), 0) # eigenvalue one-half J1 = trivial # eigenvalue one - for (eigval, eigspace) in c.operator().matrix().left_eigenspaces(): + for (eigval, eigspace) in c.operator().matrix().right_eigenspaces(): if eigval == ~(self.base_ring()(2)): J5 = eigspace else: @@ -773,105 +789,6 @@ 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: - 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. @@ -890,23 +807,42 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): True """ - return tuple( self.random_element() for idx in range(count) ) + return tuple( self.random_element() for idx in range(count) ) + + @classmethod + def random_instance(cls, field=AA, **kwargs): + """ + Return a random instance of this type of algebra. + + Beware, this will crash for "most instances" because the + constructor below looks wrong. + """ + if cls is TrivialEJA: + # The TrivialEJA class doesn't take an "n" argument because + # there's only one. + return cls(field) + n = ZZ.random_element(cls._max_test_case_size()) + 1 + return cls(n, field, **kwargs) + @cached_method def rank(self): """ Return the rank of this EJA. ALGORITHM: - The author knows of no algorithm to compute the rank of an EJA - where only the multiplication table is known. In lieu of one, we - require the rank to be specified when the algebra is created, - and simply pass along that number here. + We first compute the polynomial "column matrices" `p_{k}` that + evaluate to `x^k` on the coordinates of `x`. Then, we begin + adding them to a matrix one at a time, and trying to solve the + system that makes `p_{0}`,`p_{1}`,..., `p_{s-1}` add up to + `p_{s}`. This will succeed only when `s` is the rank of the + algebra, as proven in a recent draft paper of mine. SETUP:: - sage: from mjo.eja.eja_algebra import (JordanSpinEJA, + sage: from mjo.eja.eja_algebra import (HadamardEJA, + ....: JordanSpinEJA, ....: RealSymmetricEJA, ....: ComplexHermitianEJA, ....: QuaternionHermitianEJA, @@ -947,8 +883,80 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): sage: r > 0 or (r == 0 and J.is_trivial()) True + Ensure that computing the rank actually works, since the ranks + of all simple algebras are known and will be cached by default:: + + sage: J = HadamardEJA(4) + sage: J.rank.clear_cache() + sage: J.rank() + 4 + + :: + + sage: J = JordanSpinEJA(4) + sage: J.rank.clear_cache() + sage: J.rank() + 2 + + :: + + sage: J = RealSymmetricEJA(3) + sage: J.rank.clear_cache() + sage: J.rank() + 3 + + :: + + sage: J = ComplexHermitianEJA(2) + sage: J.rank.clear_cache() + sage: J.rank() + 2 + + :: + + sage: J = QuaternionHermitianEJA(2) + sage: J.rank.clear_cache() + sage: J.rank() + 2 + """ - return self._rank + n = self.dimension() + if n == 0: + return 0 + elif n == 1: + return 1 + + var_names = [ "X" + str(z) for z in range(1,n+1) ] + R = PolynomialRing(self.base_ring(), var_names) + vars = R.gens() + + def L_x_i_j(i,j): + # From a result in my book, these are the entries of the + # basis representation of L_x. + return sum( vars[k]*self.monomial(k).operator().matrix()[i,j] + for k in range(n) ) + + L_x = matrix(R, n, n, L_x_i_j) + x_powers = [ vars[k]*(L_x**k)*self.one().to_vector() + for k in range(n) ] + + # Can assume n >= 2 + M = matrix([x_powers[0]]) + old_rank = 1 + + for d in range(1,n): + M = matrix(M.rows() + [x_powers[d]]) + M.echelonize() + # TODO: we've basically solved the system here. + # We should save the echelonized matrix somehow + # so that it can be reused in the charpoly method. + new_rank = M.rank() + if new_rank == old_rank: + return new_rank + else: + old_rank = new_rank + + return n def vector_space(self): @@ -972,61 +980,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): Element = FiniteDimensionalEuclideanJordanAlgebraElement -class KnownRankEJA(object): - """ - A class for algebras that we actually know we can construct. The - main issue is that, for most of our methods to make sense, we need - to know the rank of our algebra. Thus we can't simply generate a - "random" algebra, or even check that a given basis and product - satisfy the axioms; because even if everything looks OK, we wouldn't - know the rank we need to actuallty build the thing. - - Not really a subclass of FDEJA because doing that causes method - resolution errors, e.g. - - TypeError: Error when calling the metaclass bases - Cannot create a consistent method resolution - order (MRO) for bases FiniteDimensionalEuclideanJordanAlgebra, - KnownRankEJA - - """ - @staticmethod - def _max_test_case_size(): - """ - Return an integer "size" that is an upper bound on the size of - this algebra when it is used in a random test - case. Unfortunately, the term "size" is quite vague -- when - dealing with `R^n` under either the Hadamard or Jordan spin - product, the "size" refers to the dimension `n`. When dealing - with a matrix algebra (real symmetric or complex/quaternion - Hermitian), it refers to the size of the matrix, which is - far less than the dimension of the underlying vector space. - - We default to five in this class, which is safe in `R^n`. The - matrix algebra subclasses (or any class where the "size" is - interpreted to be far less than the dimension) should override - with a smaller number. - """ - return 5 - - @classmethod - def random_instance(cls, field=QQ, **kwargs): - """ - Return a random instance of this type of algebra. - - Beware, this will crash for "most instances" because the - constructor below looks wrong. - """ - if cls is TrivialEJA: - # The TrivialEJA class doesn't take an "n" argument because - # there's only one. - return cls(field) - - n = ZZ.random_element(cls._max_test_case_size()) + 1 - return cls(n, field, **kwargs) - - -class HadamardEJA(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA): +class HadamardEJA(FiniteDimensionalEuclideanJordanAlgebra): """ Return the Euclidean Jordan Algebra corresponding to the set `R^n` under the Hadamard product. @@ -1066,13 +1020,14 @@ class HadamardEJA(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA): (r0, r1, r2) """ - def __init__(self, n, field=QQ, **kwargs): + 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) - return fdeja.__init__(field, mult_table, rank=n, **kwargs) + fdeja.__init__(field, mult_table, **kwargs) + self.rank.set_cache(n) def inner_product(self, x, y): """ @@ -1099,7 +1054,7 @@ class HadamardEJA(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA): return x.to_vector().inner_product(y.to_vector()) -def random_eja(field=QQ, nontrivial=False): +def random_eja(field=AA, nontrivial=False): """ Return a "random" finite-dimensional Euclidean Jordan Algebra. @@ -1113,9 +1068,13 @@ def random_eja(field=QQ, nontrivial=False): Euclidean Jordan algebra of dimension... """ - eja_classes = KnownRankEJA.__subclasses__() - if nontrivial: - eja_classes.remove(TrivialEJA) + eja_classes = [HadamardEJA, + JordanSpinEJA, + RealSymmetricEJA, + ComplexHermitianEJA, + QuaternionHermitianEJA] + if not nontrivial: + eja_classes.append(TrivialEJA) classname = choice(eja_classes) return classname.random_instance(field=field) @@ -1131,7 +1090,7 @@ class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra): # field can have dimension 4 (quaternions) too. return 2 - def __init__(self, field, basis, rank, normalize_basis=True, **kwargs): + def __init__(self, field, basis, normalize_basis=True, **kwargs): """ Compared to the superclass constructor, we take a basis instead of a multiplication table because the latter can be computed in terms @@ -1144,7 +1103,7 @@ class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra): # time to ensure that it isn't a generator expression. basis = tuple(basis) - if rank > 1 and normalize_basis: + if len(basis) > 1 and normalize_basis: # We'll need sqrt(2) to normalize the basis, and this # winds up in the multiplication table, so the whole # algebra needs to be over the field extension. @@ -1161,13 +1120,32 @@ class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra): Qs = self.multiplication_table_from_matrix_basis(basis) fdeja = super(MatrixEuclideanJordanAlgebra, self) - return fdeja.__init__(field, - Qs, - rank=rank, - natural_basis=basis, - **kwargs) + fdeja.__init__(field, Qs, natural_basis=basis, **kwargs) + return + @cached_method + def rank(self): + r""" + Override the parent method with something that tries to compute + over a faster (non-extension) field. + """ + if self._basis_normalizers is None: + # We didn't normalize, so assume that the basis we started + # with had entries in a nice field. + return super(MatrixEuclideanJordanAlgebra, self).rank() + else: + basis = ( (b/n) for (b,n) in zip(self.natural_basis(), + self._basis_normalizers) ) + + # Do this over the rationals and convert back at the end. + # Only works because we know the entries of the basis are + # integers. + J = MatrixEuclideanJordanAlgebra(QQ, + basis, + normalize_basis=False) + return J.rank() + @cached_method def _charpoly_coeff(self, i): """ @@ -1185,7 +1163,6 @@ class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra): # Do this over the rationals and convert back at the end. J = MatrixEuclideanJordanAlgebra(QQ, basis, - self.rank(), normalize_basis=False) (_,x,_,_) = J._charpoly_matrix_system() p = J._charpoly_coeff(i) @@ -1296,7 +1273,7 @@ class RealMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): return M -class RealSymmetricEJA(RealMatrixEuclideanJordanAlgebra, KnownRankEJA): +class RealSymmetricEJA(RealMatrixEuclideanJordanAlgebra): """ The rank-n simple EJA consisting of real symmetric n-by-n matrices, the usual symmetric Jordan product, and the trace inner @@ -1319,8 +1296,8 @@ class RealSymmetricEJA(RealMatrixEuclideanJordanAlgebra, KnownRankEJA): In theory, our "field" can be any subfield of the reals:: - sage: RealSymmetricEJA(2, AA) - Euclidean Jordan algebra of dimension 3 over Algebraic Real Field + sage: RealSymmetricEJA(2, RDF) + Euclidean Jordan algebra of dimension 3 over Real Double Field sage: RealSymmetricEJA(2, RR) Euclidean Jordan algebra of dimension 3 over Real Field with 53 bits of precision @@ -1412,9 +1389,10 @@ class RealSymmetricEJA(RealMatrixEuclideanJordanAlgebra, KnownRankEJA): return 4 # Dimension 10 - def __init__(self, n, field=QQ, **kwargs): + def __init__(self, n, field=AA, **kwargs): basis = self._denormalized_basis(n, field) - super(RealSymmetricEJA, self).__init__(field, basis, n, **kwargs) + super(RealSymmetricEJA, self).__init__(field, basis, **kwargs) + self.rank.set_cache(n) class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): @@ -1432,7 +1410,7 @@ class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): EXAMPLES:: - sage: F = QuadraticField(-1, 'i') + sage: F = QuadraticField(-1, 'I') sage: x1 = F(4 - 2*i) sage: x2 = F(1 + 2*i) sage: x3 = F(-i) @@ -1452,7 +1430,7 @@ class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): sage: set_random_seed() sage: n_max = ComplexMatrixEuclideanJordanAlgebra._max_test_case_size() sage: n = ZZ.random_element(n_max) - sage: F = QuadraticField(-1, 'i') + sage: F = QuadraticField(-1, 'I') sage: X = random_matrix(F, n) sage: Y = random_matrix(F, n) sage: Xe = ComplexMatrixEuclideanJordanAlgebra.real_embed(X) @@ -1495,15 +1473,15 @@ class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): ....: [ 9, 10, 11, 12], ....: [-10, 9, -12, 11] ]) sage: ComplexMatrixEuclideanJordanAlgebra.real_unembed(A) - [ 2*i + 1 4*i + 3] - [ 10*i + 9 12*i + 11] + [ 2*I + 1 4*I + 3] + [ 10*I + 9 12*I + 11] TESTS: Unembedding is the inverse of embedding:: sage: set_random_seed() - sage: F = QuadraticField(-1, 'i') + sage: F = QuadraticField(-1, 'I') sage: M = random_matrix(F, 3) sage: Me = ComplexMatrixEuclideanJordanAlgebra.real_embed(M) sage: ComplexMatrixEuclideanJordanAlgebra.real_unembed(Me) == M @@ -1521,7 +1499,12 @@ class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): field = M.base_ring() R = PolynomialRing(field, 'z') z = R.gen() - F = field.extension(z**2 + 1, 'i', embedding=CLF(-1).sqrt()) + if field is AA: + # Sage doesn't know how to embed AA into QQbar, i.e. how + # to adjoin sqrt(-1) to AA. + F = QQbar + else: + F = field.extension(z**2 + 1, 'I', embedding=CLF(-1).sqrt()) i = F.gen() # Go top-left to bottom-right (reading order), converting every @@ -1562,7 +1545,7 @@ class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): sage: Ye = y.natural_representation() sage: X = ComplexHermitianEJA.real_unembed(Xe) sage: Y = ComplexHermitianEJA.real_unembed(Ye) - sage: expected = (X*Y).trace().vector()[0] + sage: expected = (X*Y).trace().real() sage: actual = ComplexHermitianEJA.natural_inner_product(Xe,Ye) sage: actual == expected True @@ -1571,7 +1554,7 @@ class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): return RealMatrixEuclideanJordanAlgebra.natural_inner_product(X,Y)/2 -class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra, KnownRankEJA): +class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra): """ The rank-n simple EJA consisting of complex Hermitian n-by-n matrices over the real numbers, the usual symmetric Jordan product, @@ -1586,8 +1569,8 @@ class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra, KnownRankEJA): In theory, our "field" can be any subfield of the reals:: - sage: ComplexHermitianEJA(2, AA) - Euclidean Jordan algebra of dimension 4 over Algebraic Real Field + sage: ComplexHermitianEJA(2, RDF) + Euclidean Jordan algebra of dimension 4 over Real Double Field sage: ComplexHermitianEJA(2, RR) Euclidean Jordan algebra of dimension 4 over Real Field with 53 bits of precision @@ -1697,9 +1680,10 @@ class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra, KnownRankEJA): return ( s.change_ring(field) for s in S ) - def __init__(self, n, field=QQ, **kwargs): + def __init__(self, n, field=AA, **kwargs): basis = self._denormalized_basis(n,field) - super(ComplexHermitianEJA,self).__init__(field, basis, n, **kwargs) + super(ComplexHermitianEJA,self).__init__(field, basis, **kwargs) + self.rank.set_cache(n) class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): @@ -1749,7 +1733,7 @@ class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): if M.ncols() != n: raise ValueError("the matrix 'M' must be square") - F = QuadraticField(-1, 'i') + F = QuadraticField(-1, 'I') i = F.gen() blocks = [] @@ -1825,10 +1809,10 @@ class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): raise ValueError('bad on-diagonal submatrix') if submat[0,1] != -submat[1,0].conjugate(): raise ValueError('bad off-diagonal submatrix') - z = submat[0,0].vector()[0] # real part - z += submat[0,0].vector()[1]*i # imag part - z += submat[0,1].vector()[0]*j # real part - z += submat[0,1].vector()[1]*k # imag part + z = submat[0,0].real() + z += submat[0,0].imag()*i + z += submat[0,1].real()*j + z += submat[0,1].imag()*k elements.append(z) return matrix(Q, n/4, elements) @@ -1865,8 +1849,7 @@ class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): return RealMatrixEuclideanJordanAlgebra.natural_inner_product(X,Y)/4 -class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra, - KnownRankEJA): +class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra): """ The rank-n simple EJA consisting of self-adjoint n-by-n quaternion matrices, the usual symmetric Jordan product, and the @@ -1881,8 +1864,8 @@ class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra, In theory, our "field" can be any subfield of the reals:: - sage: QuaternionHermitianEJA(2, AA) - Euclidean Jordan algebra of dimension 6 over Algebraic Real Field + sage: QuaternionHermitianEJA(2, RDF) + Euclidean Jordan algebra of dimension 6 over Real Double Field sage: QuaternionHermitianEJA(2, RR) Euclidean Jordan algebra of dimension 6 over Real Field with 53 bits of precision @@ -1993,12 +1976,13 @@ class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra, return ( s.change_ring(field) for s in S ) - def __init__(self, n, field=QQ, **kwargs): + def __init__(self, n, field=AA, **kwargs): basis = self._denormalized_basis(n,field) - super(QuaternionHermitianEJA,self).__init__(field, basis, n, **kwargs) + super(QuaternionHermitianEJA,self).__init__(field, basis, **kwargs) + self.rank.set_cache(n) -class BilinearFormEJA(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA): +class BilinearFormEJA(FiniteDimensionalEuclideanJordanAlgebra): 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 = @@ -2053,7 +2037,7 @@ class BilinearFormEJA(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA): sage: actual == expected True """ - def __init__(self, n, field=QQ, B=None, **kwargs): + def __init__(self, n, field=AA, B=None, **kwargs): if B is None: self._B = matrix.identity(field, max(0,n-1)) else: @@ -2078,7 +2062,8 @@ class BilinearFormEJA(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA): # one-dimensional ambient space (because the rank is bounded # by the ambient dimension). fdeja = super(BilinearFormEJA, self) - return fdeja.__init__(field, mult_table, rank=min(n,2), **kwargs) + fdeja.__init__(field, mult_table, **kwargs) + self.rank.set_cache(min(n,2)) def inner_product(self, x, y): r""" @@ -2166,13 +2151,13 @@ class JordanSpinEJA(BilinearFormEJA): True """ - def __init__(self, n, field=QQ, **kwargs): + 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) -class TrivialEJA(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA): +class TrivialEJA(FiniteDimensionalEuclideanJordanAlgebra): """ The trivial Euclidean Jordan algebra consisting of only a zero element. @@ -2196,14 +2181,15 @@ class TrivialEJA(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA): sage: J.one().norm() 0 sage: J.one().subalgebra_generated_by() - Euclidean Jordan algebra of dimension 0 over Rational Field + Euclidean Jordan algebra of dimension 0 over Algebraic Real Field sage: J.rank() 0 """ - def __init__(self, field=QQ, **kwargs): + def __init__(self, field=AA, **kwargs): mult_table = [] fdeja = super(TrivialEJA, self) # The rank is zero using my definition, namely the dimension of the # largest subalgebra generated by any element. - return fdeja.__init__(field, mult_table, rank=0, **kwargs) + fdeja.__init__(field, mult_table, **kwargs) + self.rank.set_cache(0)