Jacobson–Morozov theorem
In mathematics, the Jacobson–Morozov theorem is the assertion that nilpotent elements in a semi-simple Lie algebra can be extended to sl2-triples. The theorem is named after , .

Statement

The statement of Jacobson–Morozov relies on the following preliminary notions: an sl2-triple in a semi-simple Lie algebra  (throughout in this article, over a field of characteristic zero) is a homomorphism of Lie algebras . Equivalently, it is a triple  of elements in  satisfying the relations 

An element  is called nilpotent, if the endomorphism  (known as the adjoint representation) is a nilpotent endomorphism. It is an elementary fact that for any sl2-triple , e must be nilpotent. The Jacobson–Morozov theorem states that, conversely, any nilpotent non-zero element  can be extended to an sl2-triple. For , the sl2-triples obtained in this way are made explicit in .

The theorem can also be stated for linear algebraic groups (again over a field k of characteristic zero): any morphism (of algebraic groups) from the additive group  to a reductive group H factors through the embedding 

Furthermore, any two such factorizations

are conjugate by a k-point of H.

Generalization
A far-reaching generalization of the theorem as formulated above can be stated as follows: the inclusion of pro-reductive groups into all linear algebraic groups, where morphisms  in both categories are taken up to conjugation by elements in , admits a left adjoint, the so-called pro-reductive envelope. This left adjoint sends the additive group  to  (which happens to be semi-simple, as opposed to pro-reductive), thereby recovering the above form of Jacobson–Morozov.
This generalized Jacobson–Morozov theorem was proven by  by appealing to methods related to Tannakian categories and by  by more geometric methods.

References

		

Lie algebras
Algebraic groups