Lie Algebras With Basis¶
AUTHORS:
Travis Scrimshaw (07-15-2013): Initial implementation
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class
sage.categories.lie_algebras_with_basis.
LieAlgebrasWithBasis
(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom_over_base_ring
Category of Lie algebras with a basis.
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class
ElementMethods
¶ Bases:
object
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lift
()¶ Lift
self
to the universal enveloping algebra.EXAMPLES:
sage: S = SymmetricGroup(3).algebra(QQ) sage: L = LieAlgebra(associative=S) sage: x = L.gen(3) sage: y = L.gen(1) sage: x.lift() b3 sage: y.lift() b1 sage: x * y b1*b3 + b4 - b5
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to_vector
(order=None)¶ Return the vector in
g.module()
corresponding to the elementself
ofg
(whereg
is the parent ofself
).Implement this if you implement
g.module()
. Seesage.categories.lie_algebras.LieAlgebras.module()
for how this is to be done.EXAMPLES:
sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example() sage: L.an_element().to_vector() (0, 0, 0)
Todo
Doctest this implementation on an example not overshadowed.
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class
ParentMethods
¶ Bases:
object
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bracket_on_basis
(x, y)¶ Return the bracket of basis elements indexed by
x
andy
wherex < y
. If this is not implemented, then the method_bracket_()
for the elements must be overwritten.EXAMPLES:
sage: L = LieAlgebras(QQ).WithBasis().example() sage: L.bracket_on_basis(Partition([3,1]), Partition([2,2,1,1])) 0
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dimension
()¶ Return the dimension of
self
.EXAMPLES:
sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example() sage: L.dimension() 3
sage: L = LieAlgebra(QQ, 'x,y', {('x','y'): {'x':1}}) sage: L.dimension() 2
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from_vector
(v, order=None)¶ Return the element of
self
corresponding to the vectorv
inself.module()
.Implement this if you implement
module()
; see the documentation ofsage.categories.lie_algebras.LieAlgebras.module()
for how this is to be done.EXAMPLES:
sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example() sage: u = L.from_vector(vector(QQ, (1, 0, 0))); u (1, 0, 0) sage: parent(u) is L True
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module
()¶ Return an \(R\)-module which is isomorphic to the underlying \(R\)-module of
self
.See
sage.categories.lie_algebras.LieAlgebras.module()
for an explanation.EXAMPLES:
sage: L = LieAlgebras(QQ).WithBasis().example() sage: L.module() Free module generated by Partitions over Rational Field
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pbw_basis
(basis_key=None, **kwds)¶ Return the Poincare-Birkhoff-Witt basis of the universal enveloping algebra corresponding to
self
.EXAMPLES:
sage: L = lie_algebras.sl(QQ, 2) sage: PBW = L.pbw_basis()
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poincare_birkhoff_witt_basis
(basis_key=None, **kwds)¶ Return the Poincare-Birkhoff-Witt basis of the universal enveloping algebra corresponding to
self
.EXAMPLES:
sage: L = lie_algebras.sl(QQ, 2) sage: PBW = L.pbw_basis()
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example
(gens=None)¶ Return an example of a Lie algebra as per
Category.example
.EXAMPLES:
sage: LieAlgebras(QQ).WithBasis().example() An example of a Lie algebra: the abelian Lie algebra on the generators indexed by Partitions over Rational Field
Another set of generators can be specified as an optional argument:
sage: LieAlgebras(QQ).WithBasis().example(Compositions()) An example of a Lie algebra: the abelian Lie algebra on the generators indexed by Compositions of non-negative integers over Rational Field
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class