what can be supported in a general Jordan Algebra.
"""
-from sage.all import *
-
-def eja_minimal_polynomial(x):
- """
- Return the minimal polynomial of ``x`` in its parent EJA
- """
- return x._x.matrix().minimal_polynomial()
+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 __init__(self, field, mult_table, names='e', category=None):
+ fda = super(FiniteDimensionalEuclideanJordanAlgebra, self)
+ fda.__init__(field, mult_table, names, category)
+
+
+ def _repr_(self):
+ """
+ Return a string representation of ``self``.
+ """
+ return "Euclidean Jordan algebra of degree {} over {}".format(self.degree(), self.base_ring())
+
+ def rank(self):
+ """
+ Return the rank of this EJA.
+ """
+ raise NotImplementedError
+
+
+ class Element(FiniteDimensionalAlgebraElement):
+ """
+ An element of a Euclidean Jordan algebra.
+
+ Since EJAs are commutative, the "right multiplication" matrix is
+ also the left multiplication matrix and must be symmetric::
+
+ sage: set_random_seed()
+ sage: J = eja_ln(5)
+ sage: J.random_element().matrix().is_symmetric()
+ True
+
+ """
+
+ def __pow__(self, n):
+ """
+ Return ``self`` raised to the power ``n``.
+
+ Jordan algebras are always power-associative; see for
+ example Faraut and Koranyi, Proposition II.1.2 (ii).
+ """
+ A = self.parent()
+ if n == 0:
+ return A.one()
+ elif n == 1:
+ return self
+ else:
+ return A.element_class(A, self.vector()*(self.matrix()**(n-1)))
+
+
+ def span_of_powers(self):
+ """
+ Return the vector space spanned by successive powers of
+ this element.
+ """
+ # The dimension of the subalgebra can't be greater than
+ # the big algebra, so just put everything into a list
+ # and let span() get rid of the excess.
+ V = self.vector().parent()
+ return V.span( (self**d).vector() for d in xrange(V.dimension()) )
+
+
+ def degree(self):
+ """
+ Compute the degree of this element the straightforward way
+ according to the definition; by appending powers to a list
+ and figuring out its dimension (that is, whether or not
+ they're linearly dependent).
+
+ EXAMPLES::
+
+ sage: J = eja_ln(4)
+ sage: J.one().degree()
+ 1
+ sage: e0,e1,e2,e3 = J.gens()
+ sage: (e0 - e1).degree()
+ 2
+
+ """
+ return self.span_of_powers().dimension()
+
+
+ def minimal_polynomial(self):
+ return self.matrix().minimal_polynomial()
+
+ def characteristic_polynomial(self):
+ return self.matrix().characteristic_polynomial()
def eja_rn(dimension, field=QQ):
# component of x; and likewise for the ith basis element e_i.
Qs = [ matrix(field, dimension, dimension, lambda k,j: 1*(k == j == i))
for i in xrange(dimension) ]
- A = FiniteDimensionalAlgebra(QQ,Qs,assume_associative=True)
- return JordanAlgebra(A)
+
+ return FiniteDimensionalEuclideanJordanAlgebra(field,Qs)
def eja_ln(dimension, field=QQ):
"""
- Return the Jordan algebra corresponding to the Lorenzt "ice cream"
+ Return the Jordan algebra corresponding to the Lorentz "ice cream"
cone of the given ``dimension``.
EXAMPLES:
Qi[0,0] = Qi[0,0] * ~field(2)
Qs.append(Qi)
- A = FiniteDimensionalAlgebra(QQ,Qs,assume_associative=True)
- return JordanAlgebra(A)
+ return FiniteDimensionalEuclideanJordanAlgebra(field,Qs)