∞-Lie theory (higher geometry)
Background
Smooth structure
Higher groupoids
Lie theory
∞-Lie groupoids
∞-Lie algebroids
Formal Lie groupoids
Cohomology
Homotopy
Related topics
Examples
-Lie groupoids
-Lie groups
-Lie algebroids
-Lie algebras
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 is semisimple precisely if the Killing form invariant polynomial
is non-degenerate as a bilinear form (hence making a metric Lie algebra).
The corresponding cocycle in Lie algebra cohomology is the one that classifies the string Lie 2-algebra-extension of .
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.)
An infinite-dimensional generalization of semisimple Lie algebras are Kac-Moody Lie algebras.
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
Representation Theory, Complex Analysis and Integral Geometry, Birkhauser (2012), 97-133, pdf
Last revised on September 2, 2024 at 11:06:30. See the history of this page for a list of all contributions to it.