semisimple Lie algebra



\infty-Lie theory

∞-Lie theory (higher geometry)


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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.


Every semisimple Lie algebra is a reductive Lie algebra.

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

x,ytr(ad xad y) \langle x,y \rangle \coloneqq 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 𝔤\mathfrak{g}.


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.)


  • 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 November 21, 2017 at 14:32:27. See the history of this page for a list of all contributions to it.