Examples of a Lie algebra with basis¶
- class sage.categories.examples.lie_algebras_with_basis.AbelianLieAlgebra(R, gens)[source]¶
Bases:
CombinatorialFreeModuleAn example of a Lie algebra: the abelian Lie algebra.
This class illustrates a minimal implementation of a Lie algebra with a distinguished basis.
- class Element[source]¶
Bases:
IndexedFreeModuleElement- lift()[source]¶
Return the lift of
selfto the universal enveloping algebra.EXAMPLES:
sage: L = LieAlgebras(QQ).WithBasis().example() sage: elt = L.an_element() sage: elt.lift() 3*P[F[2]] + 2*P[F[1]] + 2*P[F[]]
>>> from sage.all import * >>> L = LieAlgebras(QQ).WithBasis().example() >>> elt = L.an_element() >>> elt.lift() 3*P[F[2]] + 2*P[F[1]] + 2*P[F[]]
- bracket_on_basis(x, y)[source]¶
Return the Lie bracket on basis elements indexed by
xandy.EXAMPLES:
sage: L = LieAlgebras(QQ).WithBasis().example() sage: L.bracket_on_basis(Partition([4,1]), Partition([2,2,1])) 0
>>> from sage.all import * >>> L = LieAlgebras(QQ).WithBasis().example() >>> L.bracket_on_basis(Partition([Integer(4),Integer(1)]), Partition([Integer(2),Integer(2),Integer(1)])) 0
- lie_algebra_generators()[source]¶
Return the generators of
selfas a Lie algebra.EXAMPLES:
sage: L = LieAlgebras(QQ).WithBasis().example() sage: L.lie_algebra_generators() Lazy family (Term map from Partitions to An example of a Lie algebra: the abelian Lie algebra on the generators indexed by Partitions over Rational Field(i))_{i in Partitions}
>>> from sage.all import * >>> L = LieAlgebras(QQ).WithBasis().example() >>> L.lie_algebra_generators() Lazy family (Term map from Partitions to An example of a Lie algebra: the abelian Lie algebra on the generators indexed by Partitions over Rational Field(i))_{i in Partitions}
- sage.categories.examples.lie_algebras_with_basis.Example[source]¶
alias of
AbelianLieAlgebra
- class sage.categories.examples.lie_algebras_with_basis.IndexedPolynomialRing(R, indices, **kwds)[source]¶
Bases:
CombinatorialFreeModulePolynomial ring whose generators are indexed by an arbitrary set.
Todo
Currently this is just used as the universal enveloping algebra for the example of the abelian Lie algebra. This should be factored out into a more complete class.
- algebra_generators()[source]¶
Return the algebra generators of
self.EXAMPLES:
sage: L = LieAlgebras(QQ).WithBasis().example() sage: UEA = L.universal_enveloping_algebra() sage: UEA.algebra_generators() Lazy family (algebra generator map(i))_{i in Partitions}
>>> from sage.all import * >>> L = LieAlgebras(QQ).WithBasis().example() >>> UEA = L.universal_enveloping_algebra() >>> UEA.algebra_generators() Lazy family (algebra generator map(i))_{i in Partitions}
- one_basis()[source]¶
Return the index of element \(1\).
EXAMPLES:
sage: L = LieAlgebras(QQ).WithBasis().example() sage: UEA = L.universal_enveloping_algebra() sage: UEA.one_basis() 1 sage: UEA.one_basis().parent() Free abelian monoid indexed by Partitions
>>> from sage.all import * >>> L = LieAlgebras(QQ).WithBasis().example() >>> UEA = L.universal_enveloping_algebra() >>> UEA.one_basis() 1 >>> UEA.one_basis().parent() Free abelian monoid indexed by Partitions
- product_on_basis(x, y)[source]¶
Return the product of the monomials indexed by
xandy.EXAMPLES:
sage: L = LieAlgebras(QQ).WithBasis().example() sage: UEA = L.universal_enveloping_algebra() sage: I = UEA._indices sage: UEA.product_on_basis(I.an_element(), I.an_element()) P[F[]^4*F[1]^4*F[2]^6]
>>> from sage.all import * >>> L = LieAlgebras(QQ).WithBasis().example() >>> UEA = L.universal_enveloping_algebra() >>> I = UEA._indices >>> UEA.product_on_basis(I.an_element(), I.an_element()) P[F[]^4*F[1]^4*F[2]^6]