∞-Lie theory

# Contents

## Definition

A Lie algebra is semisimple if it is the direct sum of simple Lie algebras.

(Notice that this is not quite the same as a semisimple object in the category of Lie algebras, because a simple Lie algebra is not quite the same as a simple object in the LieAlg. But this is the standard terminology convention.)

By Lie integration semisimple Lie algebras correspond to Lie groups that are semisimple Lie groups.

## Properties

Every semisimple Lie algebra is a reductive Lie algebra.

A Lie algebra $\mathfrak{g}$ is semisimple precisely if the Killing form invariant polynomial

$\langle x,y \rangle := tr (ad_x \circ ad_y)$

is non-degenerate as a bilinear form.

The corresponding cocycle $\langle -,[-,-]\rangle$ in Lie algebra cohomology is the one that classifies the string Lie 2-algebra-extension of $\mathfrak{g}$.

## Classification

Since we can classify simple Lie algebras, we can classify semisimple Lie algebras; for each simple Lie algebra, we simply indicate how many times it appears in the direct-sum decomposition. (There is a theorem to prove here: that the decomposition of a semisimple Lie algebra is unique.)

An infinite-dimensional generalization of semisimple Lie algebras are Kac-Moody Lie algebras.

## References

• Robert Cahn, Semisimple Lie algebras and their representation (pdf)

Basics of the representation theory of semisimple Lie algebras is surveyed in

• Joseph Bernstein, Lectures on Lie Algebras, in: Representation Theory, Complex Analysis and Integral Geometry, Birkhauser (2012), 97-133, pdf

Revised on May 8, 2013 17:37:57 by Zoran Škoda (161.53.130.104)