Examples of a Lie algebra with basis¶
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class
sage.categories.examples.lie_algebras_with_basis.
AbelianLieAlgebra
(R, gens)¶ Bases:
sage.combinat.free_module.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.
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class
Element
¶ Bases:
sage.modules.with_basis.indexed_element.IndexedFreeModuleElement
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lift
()¶ Return the lift of
self
to 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[]]
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bracket_on_basis
(x, y)¶ Return the Lie bracket on basis elements indexed by
x
andy
.EXAMPLES:
sage: L = LieAlgebras(QQ).WithBasis().example() sage: L.bracket_on_basis(Partition([4,1]), Partition([2,2,1])) 0
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lie_algebra_generators
()¶ Return the generators of
self
as 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}
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class
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sage.categories.examples.lie_algebras_with_basis.
Example
¶ alias of
sage.categories.examples.lie_algebras_with_basis.AbelianLieAlgebra
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class
sage.categories.examples.lie_algebras_with_basis.
IndexedPolynomialRing
(R, indices, **kwds)¶ Bases:
sage.combinat.free_module.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.
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algebra_generators
()¶ 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}
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one_basis
()¶ 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
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product_on_basis
(x, y)¶ Return the product of the monomials indexed by
x
andy
.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]
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