Contents

# 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.

## Classification

Since simple Lie algebras have a classification, so do 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.)

## 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

Last revised on December 4, 2019 at 09:41:21. See the history of this page for a list of all contributions to it.