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 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 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 : any morphism 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, 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.