# nLab Jacobson-Morozov theorem

Contents

### Context

#### Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Related topics

Examples

$\infty$-Lie groupoids

$\infty$-Lie groups

$\infty$-Lie algebroids

$\infty$-Lie algebras

# Contents

## Statement

If $\mathfrak{g}$ is a semisimple Lie algebra over ground field the complex numbers, then for every nilpotent element $x \in \mathfrak{g}$ (i.e. $\underset{n \in \mathbb{N}}{\exists} [x,-]^n = 0$) there is as homomorphism of Lie algebras

$\mathfrak{sl}(2,\mathbb{C}) \longrightarrow \mathfrak{g}$

from sl(2), such that $x$ is the image of a nilpotent element of $\mathfrak{sl}(2,\mathbb{C})$.

## References

Lecture notes:

• Ana Balibanu, The Jacobson-Morozov theorem, Section 5 of“ Geometry of semisimple Lie algebras (pdf)

Original articles:

• Bertram Kostant, The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group, Amer. J. Math. 81 (1959), 973–1032 (jstor:2372999)