-from sage.matrix.constructor import matrix
-from sage.misc.cachefunc import cached_method
-from sage.rings.all import QQ
-
-from mjo.eja.eja_subalgebra import FiniteDimensionalEJASubalgebra
-
-
-class FiniteDimensionalEJAElementSubalgebra(FiniteDimensionalEJASubalgebra):
- def __init__(self, elt, **kwargs):
- superalgebra = elt.parent()
-
- # TODO: going up to the superalgebra dimension here is
- # overkill. We should append p vectors as rows to a matrix
- # and continually rref() it until the rank stops going
- # up. When n=10 but the dimension of the algebra is 1, that
- # can save a shitload of time (especially over AA).
- powers = tuple( elt**k for k in range(elt.degree()) )
-
- super().__init__(superalgebra,
- powers,
- associative=True,
- **kwargs)
-
- # The rank is the highest possible degree of a minimal
- # polynomial, and is bounded above by the dimension. We know
- # in this case that there's an element whose minimal
- # polynomial has the same degree as the space's dimension
- # (remember how we constructed the space?), so that must be
- # its rank too.
- self.rank.set_cache(self.dimension())
-
-
- @cached_method
- def one(self):
- """
- Return the multiplicative identity element of this algebra.
-
- The superclass method computes the identity element, which is
- beyond overkill in this case: the superalgebra identity
- restricted to this algebra is its identity. Note that we can't
- count on the first basis element being the identity -- it might
- have been scaled if we orthonormalized the basis.
-
- SETUP::
-
- sage: from mjo.eja.eja_algebra import (HadamardEJA,
- ....: random_eja)
-
- EXAMPLES::
-
- sage: J = HadamardEJA(5)
- sage: J.one()
- e0 + e1 + e2 + e3 + e4
- sage: x = sum(J.gens())
- sage: A = x.subalgebra_generated_by(orthonormalize=False)
- sage: A.one()
- f0
- sage: A.one().superalgebra_element()
- e0 + e1 + e2 + e3 + e4
-
- TESTS:
-
- The identity element acts like the identity over the rationals::
-
- sage: set_random_seed()
- sage: x = random_eja(field=QQ,orthonormalize=False).random_element()
- sage: A = x.subalgebra_generated_by()
- sage: x = A.random_element()
- sage: A.one()*x == x and x*A.one() == x
- True
-
- The identity element acts like the identity over the algebraic
- reals with an orthonormal basis::
-
- sage: set_random_seed()
- sage: x = random_eja().random_element()
- sage: A = x.subalgebra_generated_by()
- sage: x = A.random_element()
- sage: A.one()*x == x and x*A.one() == x
- True
-
- The matrix of the unit element's operator is the identity over
- the rationals::
-
- sage: set_random_seed()
- sage: x = random_eja(field=QQ,orthonormalize=False).random_element()
- sage: A = x.subalgebra_generated_by(orthonormalize=False)
- sage: actual = A.one().operator().matrix()
- sage: expected = matrix.identity(A.base_ring(), A.dimension())
- sage: actual == expected
- True
-
- The matrix of the unit element's operator is the identity over
- the algebraic reals with an orthonormal basis::
-
- sage: set_random_seed()
- sage: x = random_eja().random_element()
- sage: A = x.subalgebra_generated_by()
- sage: actual = A.one().operator().matrix()
- sage: expected = matrix.identity(A.base_ring(), A.dimension())
- sage: actual == expected
- True
-
- """
- if self.dimension() == 0:
- return self.zero()
-
- return self(self.superalgebra().one())
-