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Jacobson-Morozov theorem

Contents

Context

Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

Statement

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

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

from sl(2), such that xx is the image of a nilpotent element of 𝔰𝔩(2,)\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)

See also

Created on December 4, 2019 at 09:36:07. See the history of this page for a list of all contributions to it.