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semisimple Lie algebra

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 $𝔤$ is semisimple precisely if the Killing form invariant polynomial

$$⟨x,y⟩:=\mathrm{tr}({\mathrm{ad}}_x\circ {\mathrm{ad}}_y)$$\langle x,y \rangle := tr (ad_x \circ ad_y)

is non-degenerate as a bilinear form.

The corresponding cocycle $⟨-,[-,-]⟩$ in Lie algebra cohomology is the one that classifies the string Lie 2-algebra-extension of $𝔤$.

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.)

Related concepts

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