Jacobson–Morozov theorem

Statement

The statement of Jacobson–Morozov relies on the following preliminary notions: an sl2-triple in a semi-simple Lie algebra \mathfrak g (throughout in this article, over a field of characteristic zero) is a homomorphism of Lie algebras \mathfrak{sl}_2 \to \mathfrak g. Equivalently, it is a triple e, f, h of elements in \mathfrak g satisfying the relations :[h,e] = 2e, \quad [h,f] = -2f, \quad [e,f] = h. An element x \in \mathfrak g is called nilpotent, if the endomorphism [x, -] : \mathfrak g \to \mathfrak 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 \mathfrak g can be extended to an sl2-triple. For \mathfrak g = \mathfrak{sl}_n, 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 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 factorizations :SL_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.

References

References

1. {{harvtxt. Bourbaki. 2007
2. {{harvtxt. Jacobson. 1979