Lie Algebras¶
AUTHORS:
- Travis Scrimshaw (07-15-2013): Initial implementation
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sage.categories.lie_algebras.
LieAlgebras
¶ The category of Lie algebras.
EXAMPLES:
sage: C = LieAlgebras(QQ); C Category of Lie algebras over Rational Field sage: sorted(C.super_categories(), key=str) [Category of vector spaces over Rational Field]
We construct a typical parent in this category, and do some computations with it:
sage: A = C.example(); A An example of a Lie algebra: the Lie algebra from the associative algebra Symmetric group algebra of order 3 over Rational Field generated by ([2, 1, 3], [2, 3, 1]) sage: A.category() Category of Lie algebras over Rational Field sage: A.base_ring() Rational Field sage: a,b = A.lie_algebra_generators() sage: a.bracket(b) -[1, 3, 2] + [3, 2, 1] sage: b.bracket(2*a + b) 2*[1, 3, 2] - 2*[3, 2, 1] sage: A.bracket(a, b) -[1, 3, 2] + [3, 2, 1]
Please see the source code of \(A\) (with
A??
) for how to implement other Lie algebras.Todo
Many of these tests should use Lie algebras that are not the minimal example and need to be added after trac ticket #16820 (and trac ticket #16823).
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class
sage.categories.lie_algebras.
LiftMorphism
(domain, codomain)¶ Bases:
sage.categories.morphism.Morphism
The natural lifting morphism from a Lie algebra to its enveloping algebra.