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: 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
-
"""
def __pow__(self, n):
EXAMPLES:
sage: set_random_seed()
- sage: J = eja_ln(5)
- sage: x = J.random_element()
+ sage: x = random_eja().random_element()
sage: x.matrix()*x.vector() == (x**2).vector()
True
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()
+ sage: random_eja().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()
+ sage: random_eja().zero().is_nilpotent()
True
"""
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 = random_eja().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 = random_eja().random_element()
sage: x.degree() == x.minimal_polynomial().degree()
True
return elt.minimal_polynomial()
+ def quadratic_representation(self):
+ """
+ Return the quadratic representation of this element.
+
+ EXAMPLES:
+
+ The explicit form in the spin factor algebra is given by
+ Alizadeh's Example 11.12::
+
+ sage: n = ZZ.random_element(1,10).abs()
+ sage: J = eja_ln(n)
+ sage: x = J.random_element()
+ sage: x_vec = x.vector()
+ sage: x0 = x_vec[0]
+ sage: x_bar = x_vec[1:]
+ sage: A = matrix(QQ, 1, [x_vec.inner_product(x_vec)])
+ sage: B = 2*x0*x_bar.row()
+ sage: C = 2*x0*x_bar.column()
+ sage: D = identity_matrix(QQ, n-1)
+ sage: D = (x0^2 - x_bar.inner_product(x_bar))*D
+ sage: D = D + 2*x_bar.tensor_product(x_bar)
+ sage: Q = block_matrix(2,2,[A,B,C,D])
+ sage: Q == x.quadratic_representation()
+ True
+
+ """
+ return 2*(self.matrix()**2) - (self**2).matrix()
+
+
def span_of_powers(self):
"""
Return the vector space spanned by successive powers of
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 = random_eja().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: set_random_seed()
+ sage: x = random_eja().random_element()
sage: u = x.subalgebra_generated_by().random_element()
sage: u.matrix()*u.vector() == (u**2).vector()
True
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)