Jacobson–Morozov theorem explained

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

ak g (throughout in this article, over a field of characteristic zero) is a homomorphism of Lie algebras ak{sl} _{2} \to ak g . Equivalently, it is a triple e, f, h of elements in ak g satisfying the relations [h,e] = 2 e, [h,f] = -2 f, [e,f] = h. An element x \in ak g is called nilpotent, if the endomorphism [x, - ] : ak g \to ak g (known as the adjoint representation) is a nilpotent endomorphism. It is an elementary fact that for any sl₂-triple (e, f, h) , e must be nilpotent. The Jacobson - Morozov theorem states that, conversely, any nilpotent non-zero element e \in ak g can be extended to an sl₂-triple. For ak g = ak{sl} _{n} , the sl₂-triples obtained in this way are made explicit in .G _{a} to a reductive group H factors through the embedding G _{a} \to SL _{2,} x \mapsto \left ( \begin{array}{cc} 1 & x \ 0 & 1 \end{array} \right ). Furthermore, any two such factorizationsSL _{2} \to H 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

G \to H in both categories are taken up to conjugation by elements in H(k) , admits a left adjoint, the so-called pro-reductive envelope. This left adjoint sends the additive group G _{a} to SL _{2} (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