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):
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
+
+ """
+ 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!
+ subalg_self = assoc_subalg(V.coordinates(self.vector()))
+ return subalg_self.matrix().minimal_polynomial()
+
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)
+ return FiniteDimensionalEuclideanJordanAlgebra(field,Qs,rank=2)