nLab 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 $\mathfrak{g}$ is semisimple precisely if the Killing form invariant polynomial

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

  is non-degenerate as a bilinear form (hence making $\mathfrak{g}$ a metric Lie algebra).

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

• Jacobson-Morozov theorem

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.)

Related concepts

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

• exceptional Lie algebra

References

• Jean-Pierre Serre: Semisimple Lie algebras, Chapter VI in: Lie Algebras and Lie Groups – 1964 Lectures given at Harvard University, Lecture Notes in Mathematics 1500, Springer (1992) [doi:10.1007/978-3-540-70634-2]

• 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