what can be supported in a general Jordan Algebra.
"""
+from sage.categories.magmatic_algebras import MagmaticAlgebras
+from sage.structure.element import is_Matrix
+from sage.structure.category_object import normalize_names
+
from sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra import FiniteDimensionalAlgebra
from sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra_element import FiniteDimensionalAlgebraElement
class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
@staticmethod
- def __classcall__(cls, field, mult_table, names='e', category=None):
- fda = super(FiniteDimensionalEuclideanJordanAlgebra, cls)
- return fda.__classcall_private__(cls,
- field,
- mult_table,
- names,
- category)
+ def __classcall_private__(cls,
+ field,
+ mult_table,
+ names='e',
+ assume_associative=False,
+ category=None,
+ rank=None):
+ n = len(mult_table)
+ mult_table = [b.base_extend(field) for b in mult_table]
+ for b in mult_table:
+ b.set_immutable()
+ if not (is_Matrix(b) and b.dimensions() == (n, n)):
+ raise ValueError("input is not a multiplication table")
+ mult_table = tuple(mult_table)
+
+ cat = MagmaticAlgebras(field).FiniteDimensional().WithBasis()
+ cat.or_subcategory(category)
+ if assume_associative:
+ cat = cat.Associative()
+
+ names = normalize_names(n, names)
- def __init__(self, field, mult_table, names='e', category=None):
+ fda = super(FiniteDimensionalEuclideanJordanAlgebra, cls)
+ return fda.__classcall__(cls,
+ field,
+ mult_table,
+ assume_associative=assume_associative,
+ names=names,
+ category=cat,
+ rank=rank)
+
+
+ def __init__(self, field,
+ mult_table,
+ names='e',
+ assume_associative=False,
+ category=None,
+ rank=None):
+ self._rank = rank
fda = super(FiniteDimensionalEuclideanJordanAlgebra, self)
- fda.__init__(field, mult_table, names, category)
+ fda.__init__(field,
+ mult_table,
+ names=names,
+ category=category)
def _repr_(self):
"""
Return a string representation of ``self``.
"""
- return "Euclidean Jordan algebra of degree {} over {}".format(self.degree(), self.base_ring())
+ fmt = "Euclidean Jordan algebra of degree {} over {}"
+ return fmt.format(self.degree(), self.base_ring())
def rank(self):
"""
Return the rank of this EJA.
"""
- raise NotImplementedError
+ if self._rank is None:
+ raise ValueError("no rank specified at genesis")
+ else:
+ return self._rank
class Element(FiniteDimensionalAlgebraElement):
also the left multiplication matrix and must be symmetric::
sage: set_random_seed()
+ sage: n = ZZ.random_element(1,10).abs()
+ sage: J = eja_rn(5)
+ sage: J.random_element().matrix().is_symmetric()
+ True
sage: J = eja_ln(5)
sage: J.random_element().matrix().is_symmetric()
True
Jordan algebras are always power-associative; see for
example Faraut and Koranyi, Proposition II.1.2 (ii).
+
+ .. WARNING:
+
+ We have to override this because our superclass uses row vectors
+ instead of column vectors! We, on the other hand, assume column
+ vectors everywhere.
+
+ EXAMPLES:
+
+ sage: set_random_seed()
+ sage: J = eja_ln(5)
+ sage: x = J.random_element()
+ sage: x.matrix()*x.vector() == (x**2).vector()
+ True
+
"""
A = self.parent()
if n == 0:
elif n == 1:
return self
else:
- return A.element_class(A, self.vector()*(self.matrix()**(n-1)))
+ return A.element_class(A, (self.matrix()**(n-1))*self.vector())
+
+
+ def is_regular(self):
+ """
+ Return whether or not this is a regular element.
+
+ EXAMPLES:
+
+ The identity element always has degree one, but any element
+ linearly-independent from it is regular::
+ sage: J = eja_ln(5)
+ sage: J.one().is_regular()
+ False
+ sage: e0, e1, e2, e3, e4 = J.gens() # e0 is the identity
+ sage: for x in J.gens():
+ ....: (J.one() + x).is_regular()
+ False
+ True
+ True
+ True
+ True
+
+ """
+ return self.degree() == self.parent().rank()
def span_of_powers(self):
"""
sage: (e0 - e1).degree()
2
+ In the spin factor algebra (of rank two), all elements that
+ aren't multiples of the identity are regular::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(1,10).abs()
+ sage: J = eja_ln(n)
+ sage: x = J.random_element()
+ sage: x == x.coefficient(0)*J.one() or x.degree() == 2
+ True
+
"""
return self.span_of_powers().dimension()
+ def matrix(self):
+ """
+ Return the matrix that represents left- (or right-)
+ multiplication by this element in the parent algebra.
+
+ We have to override this because the superclass method
+ returns a matrix that acts on row vectors (that is, on
+ the right).
+ """
+ fda_elt = FiniteDimensionalAlgebraElement(self.parent(), self)
+ return fda_elt.matrix().transpose()
+
+
+ def subalgebra_generated_by(self):
+ """
+ Return the associative subalgebra of the parent EJA generated
+ by this element.
+
+ TESTS::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(1,10).abs()
+ sage: J = eja_rn(n)
+ sage: x = J.random_element()
+ sage: x.subalgebra_generated_by().is_associative()
+ True
+ sage: J = eja_ln(n)
+ sage: x = J.random_element()
+ sage: x.subalgebra_generated_by().is_associative()
+ True
+
+ Squaring in the subalgebra should be the same thing as
+ squaring in the superalgebra::
+
+ sage: J = eja_ln(5)
+ sage: x = J.random_element()
+ sage: u = x.subalgebra_generated_by().random_element()
+ sage: u.matrix()*u.vector() == (u**2).vector()
+ True
+
+ """
+ # First get the subspace spanned by the powers of myself...
+ V = self.span_of_powers()
+ F = self.base_ring()
+
+ # Now figure out the entries of the right-multiplication
+ # matrix for the successive basis elements b0, b1,... of
+ # that subspace.
+ mats = []
+ for b_right in V.basis():
+ eja_b_right = self.parent()(b_right)
+ b_right_rows = []
+ # The first row of the right-multiplication matrix by
+ # b1 is what we get if we apply that matrix to b1. The
+ # second row of the right multiplication matrix by b1
+ # is what we get when we apply that matrix to b2...
+ #
+ # IMPORTANT: this assumes that all vectors are COLUMN
+ # vectors, unlike our superclass (which uses row vectors).
+ for b_left in V.basis():
+ eja_b_left = self.parent()(b_left)
+ # Multiply in the original EJA, but then get the
+ # coordinates from the subalgebra in terms of its
+ # basis.
+ this_row = V.coordinates((eja_b_left*eja_b_right).vector())
+ b_right_rows.append(this_row)
+ b_right_matrix = matrix(F, b_right_rows)
+ mats.append(b_right_matrix)
+
+ # It's an algebra of polynomials in one element, and EJAs
+ # are power-associative.
+ #
+ # TODO: choose generator names intelligently.
+ return FiniteDimensionalEuclideanJordanAlgebra(F, mats, assume_associative=True, names='f')
+
+
def minimal_polynomial(self):
- return self.matrix().minimal_polynomial()
+ """
+ EXAMPLES::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(1,10).abs()
+ sage: J = eja_rn(n)
+ sage: x = J.random_element()
+ sage: x.degree() == x.minimal_polynomial().degree()
+ True
+
+ ::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(1,10).abs()
+ sage: J = eja_ln(n)
+ sage: x = J.random_element()
+ sage: x.degree() == x.minimal_polynomial().degree()
+ True
+
+ The minimal polynomial and the characteristic polynomial coincide
+ and are known (see Alizadeh, Example 11.11) for all elements of
+ the spin factor algebra that aren't scalar multiples of the
+ identity::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(2,10).abs()
+ sage: J = eja_ln(n)
+ sage: y = J.random_element()
+ sage: while y == y.coefficient(0)*J.one():
+ ....: y = J.random_element()
+ sage: y0 = y.vector()[0]
+ sage: y_bar = y.vector()[1:]
+ sage: actual = y.minimal_polynomial()
+ sage: x = SR.symbol('x', domain='real')
+ sage: expected = x^2 - 2*y0*x + (y0^2 - norm(y_bar)^2)
+ sage: bool(actual == expected)
+ True
+
+ """
+ # The element we're going to call "minimal_polynomial()" on.
+ # Either myself, interpreted as an element of a finite-
+ # dimensional algebra, or an element of an associative
+ # subalgebra.
+ elt = None
+
+ if self.parent().is_associative():
+ elt = FiniteDimensionalAlgebraElement(self.parent(), self)
+ else:
+ V = self.span_of_powers()
+ assoc_subalg = self.subalgebra_generated_by()
+ # Mis-design warning: the basis used for span_of_powers()
+ # and subalgebra_generated_by() must be the same, and in
+ # the same order!
+ elt = assoc_subalg(V.coordinates(self.vector()))
+
+ # Recursive call, but should work since elt lives in an
+ # associative algebra.
+ return elt.minimal_polynomial()
+
+
+ def is_nilpotent(self):
+ """
+ Return whether or not some power of this element is zero.
+
+ The superclass method won't work unless we're in an
+ associative algebra, and we aren't. However, we generate
+ an assocoative subalgebra and we're nilpotent there if and
+ only if we're nilpotent here (probably).
+
+ TESTS:
+
+ The identity element is never nilpotent::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(2,10).abs()
+ sage: J = eja_rn(n)
+ sage: J.one().is_nilpotent()
+ False
+ sage: J = eja_ln(n)
+ sage: J.one().is_nilpotent()
+ False
+
+ The additive identity is always nilpotent::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(2,10).abs()
+ sage: J = eja_rn(n)
+ sage: J.zero().is_nilpotent()
+ True
+ sage: J = eja_ln(n)
+ sage: J.zero().is_nilpotent()
+ True
+
+ """
+ # The element we're going to call "is_nilpotent()" on.
+ # Either myself, interpreted as an element of a finite-
+ # dimensional algebra, or an element of an associative
+ # subalgebra.
+ elt = None
+
+ if self.parent().is_associative():
+ elt = FiniteDimensionalAlgebraElement(self.parent(), self)
+ else:
+ V = self.span_of_powers()
+ assoc_subalg = self.subalgebra_generated_by()
+ # Mis-design warning: the basis used for span_of_powers()
+ # and subalgebra_generated_by() must be the same, and in
+ # the same order!
+ elt = assoc_subalg(V.coordinates(self.vector()))
+
+ # Recursive call, but should work since elt lives in an
+ # associative algebra.
+ return elt.is_nilpotent()
+
+
+ def subalgebra_idempotent(self):
+ """
+ Find an idempotent in the associative subalgebra I generate
+ using Proposition 2.3.5 in Baes.
+
+ TESTS::
+
+ sage: set_random_seed()
+ sage: J = eja_rn(5)
+ sage: c = J.random_element().subalgebra_idempotent()
+ sage: c^2 == c
+ True
+ sage: J = eja_ln(5)
+ sage: c = J.random_element().subalgebra_idempotent()
+ sage: c^2 == c
+ True
+
+ """
+ if self.is_nilpotent():
+ raise ValueError("this only works with non-nilpotent elements!")
+
+ V = self.span_of_powers()
+ J = self.subalgebra_generated_by()
+ # Mis-design warning: the basis used for span_of_powers()
+ # and subalgebra_generated_by() must be the same, and in
+ # the same order!
+ u = J(V.coordinates(self.vector()))
+
+ # The image of the matrix of left-u^m-multiplication
+ # will be minimal for some natural number s...
+ s = 0
+ minimal_dim = V.dimension()
+ for i in xrange(1, V.dimension()):
+ this_dim = (u**i).matrix().image().dimension()
+ if this_dim < minimal_dim:
+ minimal_dim = this_dim
+ s = i
+
+ # Now minimal_matrix should correspond to the smallest
+ # non-zero subspace in Baes's (or really, Koecher's)
+ # proposition.
+ #
+ # However, we need to restrict the matrix to work on the
+ # subspace... or do we? Can't we just solve, knowing that
+ # A(c) = u^(s+1) should have a solution in the big space,
+ # too?
+ #
+ # Beware, solve_right() means that we're using COLUMN vectors.
+ # Our FiniteDimensionalAlgebraElement superclass uses rows.
+ u_next = u**(s+1)
+ A = u_next.matrix()
+ c_coordinates = A.solve_right(u_next.vector())
+
+ # Now c_coordinates is the idempotent we want, but it's in
+ # the coordinate system of the subalgebra.
+ #
+ # We need the basis for J, but as elements of the parent algebra.
+ #
+ basis = [self.parent(v) for v in V.basis()]
+ return self.parent().linear_combination(zip(c_coordinates, basis))
+
+
def characteristic_polynomial(self):
return self.matrix().characteristic_polynomial()
Qs = [ matrix(field, dimension, dimension, lambda k,j: 1*(k == j == i))
for i in xrange(dimension) ]
- return FiniteDimensionalEuclideanJordanAlgebra(field,Qs)
+ return FiniteDimensionalEuclideanJordanAlgebra(field,Qs,rank=dimension)
def eja_ln(dimension, field=QQ):
Qi[0,0] = Qi[0,0] * ~field(2)
Qs.append(Qi)
- return FiniteDimensionalEuclideanJordanAlgebra(field,Qs)
+ # The rank of the spin factor algebra is two, UNLESS we're in a
+ # one-dimensional ambient space (the rank is bounded by the
+ # ambient dimension).
+ rank = min(dimension,2)
+ return FiniteDimensionalEuclideanJordanAlgebra(field,Qs,rank=rank)
projects / sage.d.git / blobdiff