nLab
semisimple Lie algebra

Contents

Context

\infty-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

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

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.