Examples of a Lie algebra with basis

class sage.categories.examples.lie_algebras_with_basis.AbelianLieAlgebra(R, gens)

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

Bases: IndexedFreeModuleElement

lift()

Return the lift of self to the universal enveloping algebra.

EXAMPLES:

Sage

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[]]

Python

>>> 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)

Return the Lie bracket on basis elements indexed by x and y.

EXAMPLES:

Sage

sage: L = LieAlgebras(QQ).WithBasis().example()
sage: L.bracket_on_basis(Partition([4,1]), Partition([2,2,1]))
0

Python

>>> 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()

Return the generators of self as a Lie algebra.

EXAMPLES:

Sage

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}

Python

>>> 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

alias of AbelianLieAlgebra

class sage.categories.examples.lie_algebras_with_basis.IndexedPolynomialRing(R, indices, **kwds)

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()

Return the algebra generators of self.

EXAMPLES:

Sage

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}

Python

>>> 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()

Return the index of element \(1\).

EXAMPLES:

Sage

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

Python

>>> 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)

Return the product of the monomials indexed by x and y.

EXAMPLES:

Sage

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]

Python

>>> 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]