Squaring in the subalgebra should be the same thing as
squaring in the superalgebra::
+ sage: set_random_seed()
sage: J = eja_ln(5)
sage: x = J.random_element()
sage: u = x.subalgebra_generated_by().random_element()
def mat2vec(m):
return vector(field, m.list())
+ def vec2mat(v):
+ return matrix(field, dimension, v.list())
+
W = V.span( mat2vec(s) for s in S )
+ # Taking the span above reorders our basis (thanks, jerk!) so we
+ # need to put our "matrix basis" in the same order as the
+ # (reordered) vector basis.
+ S = [ vec2mat(b) for b in W.basis() ]
+
for s in S:
# Brute force the multiplication-by-s matrix by looping
# through all elements of the basis and doing the computation
Qs.append(Q)
return FiniteDimensionalEuclideanJordanAlgebra(field,Qs,rank=dimension)
+
+
+def random_eja():
+ """
+ Return a "random" finite-dimensional Euclidean Jordan Algebra.
+
+ ALGORITHM:
+
+ For now, we choose a random natural number ``n`` (greater than zero)
+ and then give you back one of the following:
+
+ * The cartesian product of the rational numbers ``n`` times; this is
+ ``QQ^n`` with the Hadamard product.
+
+ * The Jordan spin algebra on ``QQ^n``.
+
+ * The ``n``-by-``n`` rational symmetric matrices with the symmetric
+ product.
+
+ Later this might be extended to return Cartesian products of the
+ EJAs above.
+
+ TESTS::
+
+ sage: random_eja()
+ Euclidean Jordan algebra of degree...
+
+ """
+ n = ZZ.random_element(1,10).abs()
+ constructor = choice([eja_rn, eja_ln, eja_sn])
+ return constructor(dimension=n, field=QQ)