Lie algebra cohomology
Encyclopedia : L : LI : LIE : Lie algebra cohomology
Lie algebra cohomology is a cohomology theory for Lie algebras. It was defined algebraically in a 1948 paper of Claude Chevalley and Samuel Eilenberg, entitled Cohomology theory of Lie groups and Lie algebras. It was used as a tool to study the cohomology of the underlying topological spaces of Lie groups. In the paper above, a specific complex now called the Chevalley-Eilenberg complex, or also the Koszul complex, is defined for a module over a Lie algebra, and its cohomology is taken in the normal sense.
The foundations can also be laid out using an equivalence of categories. The category of modules over a given Lie algebra is equivalent to the category of modules over its universal enveloping algebra, a (usually non-commutative) ring. Then the cohomology of a module over a Lie algebra can be defined via this equivalence.
Some important basic results about the cohomology of Lie algebras include Whitehead's lemmas, Weyl's theorem, and the Levi decomposition theorem.
See also: BRST formalism in theoretical physics.
References
- Chevalley, C; Eilenberg, S. Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc. 63 (1948), 85-124.
- Knapp, A. W. Lie groups, Lie algebras and cohomology (1988). Princeton University Press.
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