Examples of a Lie algebra with basis¶
- class sage.categories.examples.lie_algebras_with_basis.AbelianLieAlgebra(R, gens)[source]¶
- Bases: - CombinatorialFreeModule- An 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 - xand- y.- 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: - CombinatorialFreeModule- Polynomial 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 - xand- y.- 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]