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
def __init__(self,
field,
mult_table,
- rank,
prefix='e',
category=None,
natural_basis=None,
# a real embedding.
raise ValueError('field is not real')
- self._rank = rank
self._natural_basis = natural_basis
if category is None:
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
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]
)
::
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.
True
"""
- return tuple( self.random_element() for idx in range(count) )
-
+ return tuple( self.random_element() for idx in range(count) )
+ @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,
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):
return 5
@classmethod
- def random_instance(cls, field=QQ, **kwargs):
+ def random_instance(cls, field=AA, **kwargs):
"""
Return a random instance of this type of algebra.
(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):
"""
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.
# 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
# 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.
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):
# 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)
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
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):
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)
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)
....: [ 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
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
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
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
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):
if M.ncols() != n:
raise ValueError("the matrix 'M' must be square")
- F = QuadraticField(-1, 'i')
+ F = QuadraticField(-1, 'I')
i = F.gen()
blocks = []
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)
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
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):
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:
# 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"""
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)
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)
projects / sage.d.git / blobdiff