Jacobson–Morozov theorem
# 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 $sl_2-triples.$ The theorem is named after Jacobson 1951, Morozov 1942.

## Statement

The statement of Jacobson–Morozov relies on the following preliminary notions: an $sl_2-triple$ 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 $sl_2-triple$ $(e,f,h)$, 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 $sl_2-triple.$ For ${\mathfrak {g}}={\mathfrak {sl}}_{n}$, the $sl_2-triples$ obtained in this way are made explicit in Chriss & Ginzburg (1997, p. 184).

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 André & Kahn (2002, Theorem 19.3.1) by appealing to methods related to Tannakian categories and by O'Sullivan (2010) by more geometric methods.

