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 $$\mathfrak g$$ (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.