Lie algebra cohomology

From Wikipedia, the free encyclopedia

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.

In particular, let $\mathfrak g$ be a Lie algebra over a commutative ring R with universal enveloping algebra $U\mathfrak g$ , and let M be a representation of $\mathfrak g$ (equivalently, a $U\mathfrak g$ -module). Considering R as a trivial representation of $\mathfrak g$ , one defines the cohomology groups

$\mathrm{H}^n(\mathfrak{g}; M) := \mathrm{Ext}^n_{U\mathfrak{g}}(R, M)$

(see Ext functor for the definition of Ext). Equivalently, these are the right derived functors of the left exact invariant submodule functor

$M \mapsto M^{\mathfrak{g}} := \{ m \in M \mid gm = 0\ \text{ for all } g \in \mathfrak{g}\}$ .

Analogously, one can define Lie algebra homology as

$\mathrm{H}_n(\mathfrak{g}; M) := \mathrm{Tor}_n^{U\mathfrak{g}}(R, M)$

(see Tor functor for the definition of Tor), which is equivalent to the left derived functors of the right exact coinvariants functor

$M \mapsto M_{\mathfrak{g}} := M / \mathfrak{g} M$ .

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.

[edit] References

[1] 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.

Lie algebra cohomology

From Wikipedia, the free encyclopedia

[edit] References

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