what can be supported in a general Jordan Algebra.
"""
-from sage.all import *
+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.
+ """
+
+ 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 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
+
+ """
+ d = 0
+ V = self.vector().parent()
+ vectors = [(self**d).vector()]
+ while V.span(vectors).dimension() > d:
+ d += 1
+ vectors.append((self**d).vector())
+ return d
+
+ def minimal_polynomial(self):
+ return self.matrix().minimal_polynomial()
+
+ def characteristic_polynomial(self):
+ return self.matrix().characteristic_polynomial()
+
def eja_rn(dimension, field=QQ):
"""
Return the Euclidean Jordan Algebra corresponding to the set
`R^n` under the Hadamard product.
+
+ EXAMPLES:
+
+ This multiplication table can be verified by hand::
+
+ sage: J = eja_rn(3)
+ sage: e0,e1,e2 = J.gens()
+ sage: e0*e0
+ e0
+ sage: e0*e1
+ 0
+ sage: e0*e2
+ 0
+ sage: e1*e1
+ e1
+ sage: e1*e2
+ 0
+ sage: e2*e2
+ e2
+
"""
# The FiniteDimensionalAlgebra constructor takes a list of
# matrices, the ith representing right multiplication by the ith
# 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 Lorentz "ice cream"
+ cone of the given ``dimension``.
+
+ EXAMPLES:
+
+ This multiplication table can be verified by hand::
+
+ sage: J = eja_ln(4)
+ sage: e0,e1,e2,e3 = J.gens()
+ sage: e0*e0
+ e0
+ sage: e0*e1
+ e1
+ sage: e0*e2
+ e2
+ sage: e0*e3
+ e3
+ sage: e1*e2
+ 0
+ sage: e1*e3
+ 0
+ sage: e2*e3
+ 0
+
+ In one dimension, this is the reals under multiplication::
+
+ sage: J1 = eja_ln(1)
+ sage: J2 = eja_rn(1)
+ sage: J1 == J2
+ True
+
+ """
+ Qs = []
+ id_matrix = identity_matrix(field,dimension)
+ for i in xrange(dimension):
+ ei = id_matrix.column(i)
+ Qi = zero_matrix(field,dimension)
+ Qi.set_row(0, ei)
+ Qi.set_column(0, ei)
+ Qi += diagonal_matrix(dimension, [ei[0]]*dimension)
+ # The addition of the diagonal matrix adds an extra ei[0] in the
+ # upper-left corner of the matrix.
+ Qi[0,0] = Qi[0,0] * ~field(2)
+ Qs.append(Qi)
+
+ return FiniteDimensionalEuclideanJordanAlgebra(field,Qs)