natural_basis=natural_basis)
+ def _a_regular_element(self):
+ """
+ Override the superalgebra method to return the one
+ regular element that is sure to exist in this
+ subalgebra, namely the element that generated it.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import random_eja
+
+ TESTS::
+
+ sage: set_random_seed()
+ sage: J = random_eja().random_element().subalgebra_generated_by()
+ sage: J._a_regular_element().is_regular()
+ True
+
+ """
+ if self.dimension() == 0:
+ return self.zero()
+ else:
+ return self.monomial(1)
+
+
def _element_constructor_(self, elt):
"""
Construct an element of this subalgebra from the given one.
return self.monomial(self.one_basis())
+ def natural_basis_space(self):
+ """
+ Return the natural basis space of this algebra, which is identical
+ to that of its superalgebra.
+
+ This is correct "by definition," and avoids a mismatch when the
+ subalgebra is trivial (with no natural basis to infer anything
+ from) and the parent is not.
+ """
+ return self.superalgebra().natural_basis_space()
+
+
def superalgebra(self):
"""
Return the superalgebra that this algebra was generated from.
EXAMPLES::
sage: J = RealSymmetricEJA(3)
- sage: x = sum( i*J.gens()[i] for i in range(6) )
+ sage: x = J.monomial(0) + 2*J.monomial(2) + 5*J.monomial(5)
sage: K = FiniteDimensionalEuclideanJordanElementSubalgebra(x)
sage: K.vector_space()
- Vector space of degree 6 and dimension 3 over Rational Field
+ Vector space of degree 6 and dimension 3 over...
User basis matrix:
[ 1 0 1 0 0 1]
- [ 0 1 2 3 4 5]
- [10 14 21 19 31 50]
+ [ 1 0 2 0 0 5]
+ [ 1 0 4 0 0 25]
sage: (x^0).to_vector()
(1, 0, 1, 0, 0, 1)
sage: (x^1).to_vector()
- (0, 1, 2, 3, 4, 5)
+ (1, 0, 2, 0, 0, 5)
sage: (x^2).to_vector()
- (10, 14, 21, 19, 31, 50)
+ (1, 0, 4, 0, 0, 25)
"""
return self._vector_space