Lie Algebras

AUTHORS:

  • Travis Scrimshaw (07-15-2013): Initial implementation
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).

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.