Lie algebra cohomology
In mathematics, Lie algebra cohomology is a cohomology theory for Lie algebras. It was defined by Claude Chevalley and Samuel Eilenberg (1948) in order to give an algebraic construction of the cohomology of the underlying topological spaces of compact Lie groups. In the paper above, a specific chain complex, called the Koszul complex, is defined for a module over a Lie algebra, and its cohomology is taken in the normal sense.
If is a compact simply connected Lie group, then it is determined by its Lie algebra, so it should be possible to calculate its cohomology from the Lie algebra. This can be done as follows. Its cohomology is the de Rham cohomology of the complex of differential forms on . Using an averaging process, this complex can be replaced by the complex of left-invariant differential forms. The left-invariant forms, meanwhile, are determined by their values at the identity, so that the space of left-invariant differential forms can be identified with the exterior algebra of the Lie algebra, with a suitable differential.
The construction of this differential on an exterior algebra makes sense for any Lie algebra, so is used to define Lie algebra cohomology for all Lie algebras. More generally one uses a similar construction to define Lie algebra cohomology with coefficients in a module.
It should be noted that if is a simply connected noncompact Lie group, the Lie algebra cohomology of the associated Lie algebra does not necessarily reproduce the de Rham cohomology of . The reason for this is that the passage from the complex of all differential forms to the complex of left-invariant differential forms uses an averaging process that only makes sense for compact groups.
Let be a Lie algebra over a commutative ring R with universal enveloping algebra , and let M be a representation of (equivalently, a -module). Considering R as a trivial representation of , one defines the cohomology groups
Analogously, one can define Lie algebra homology as
The Lie algebra cohomology of the Lie algebra over the field , with values in the left -module can be computed using the Chevalley-Eilenberg complex . The -cochains in this complex are the alternating -multilinear functions of variables with values in . The coboundary of an -cochain is the -cochain given by
where the caret signifies omitting that argument.
Cohomology in small dimensions
The zeroth cohomology group is (by definition) the invariants of the Lie algebra acting on the module:
The first cohomology group is the space Der of derivations modulo the space Ider of inner derivations
where a derivation is a map d from the Lie algebra to M such that
and is called inner if it is given by
for some a in M.
The second cohomology group
is the space of equivalence classes of Lie algebra extensions
of the Lie algebra by the module M.
There do not seem to be any similar easy interpretations for the higher cohomology groups.
- BRST formalism in theoretical physics.
- Chevalley, Claude; Eilenberg, Samuel (1948), "Cohomology Theory of Lie Groups and Lie Algebras", Transactions of the American Mathematical Society, Providence, R.I.: American Mathematical Society, 63 (1): 85–124, doi:10.2307/1990637, ISSN 0002-9947, JSTOR 1990637, MR 0024908
- Hilton, Peter J.; Stammbach, Urs (1997), A course in homological algebra, Graduate Texts in Mathematics, 4 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-94823-2, MR 1438546
- Knapp, Anthony W. (1988), Lie groups, Lie algebras, and cohomology, Mathematical Notes, 34, Princeton University Press, ISBN 978-0-691-08498-5, MR 0938524
- Weibel, Charles A. (1994). An introduction to homological algebra. Cambridge University Press. p. 240.