Lie algebra cohomology
In mathematics, Lie algebra cohomology is a cohomology theory for Lie algebras. It was first introduced in 1929 by Élie Cartan to study the topology of Lie groups and homogeneous spaces by relating cohomological methods of Georges de Rham to properties of the Lie algebra. It was later extended by Claude Chevalley and Samuel Eilenberg (1948) to coefficients in an arbitrary Lie module.
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 it 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.
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
Let be a Lie algebra over a field , with a left action on the -module . The elements of the Chevalley–Eilenberg complex
are called cochains from to . A homogeneous -cochain from to is thus an alternating -multilinear function . The Chevalley–Eilenberg complex is canonically isomorphic to the tensor product , where denotes the dual vector space of .
The Lie bracket on induces a transpose application by duality. The latter is sufficient to define a derivation of the complex of cochains from to by extending according to the graded Leibniz rule. It follows from the Jacobi identity that satisfies and is in fact a differential. In this setting, is viewed as a trivial -module while may be thought of as constants.
In general, let denote the left action of on and regard it as an application . The Chevalley–Eilenberg differential is then the unique derivation extending and according to the graded Leibniz rule, the nilpotency condition following from the Lie algebra homomorphism from to and the Jacobi identity in .
Explicitly, the differential of the -cochain is the -cochain given by:
where the caret signifies omitting that argument.
When is a real Lie group with Lie algebra , the Chevalley–Eilenberg complex may also be canonically identified with the space of left-invariant forms with values in , denoted by . The Chevalley–Eilenberg differential may then be thought of as a restriction of the covariant derivative on the trivial fiber bundle , equipped with the equivariant connection associated with the left action of on . In the particular case where is equipped with the trivial action of , the Chevalley–Eilenberg differential coincides with the restriction of the de Rham differential on to the subspace of left-invariant differential forms.
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 from the Lie algebra to such that
and is called inner if it is given by
for some in .
The second cohomology group
is the space of equivalence classes of Lie algebra extensions
of the Lie algebra by the module .
Similarly, any element of the cohomology group gives an equivalence class of ways to extend the Lie algebra to a "Lie -algebra" with in grade zero and in grade . A Lie -algebra is a homotopy Lie algebra with nonzero terms only in degrees 0 through .
- Cartan, Élie (1929). "Sur les invariants intégraux de certains espaces homogènes clos". Annales de la Société Polonaise de Mathématique. 8: 181–225.
- Koszul, Jean-Louis (1950). "Homologie et cohomologie des algèbres de Lie". Bulletin de la Société Mathématique de France. 78: 65–127. Archived from the original on 2019-04-21. Retrieved 2019-05-03.
- Weibel, Charles A. (1994). An introduction to homological algebra. Cambridge University Press. p. 240.
- Baez, John C.; Crans, Alissa S. (2004). "Higher-dimensional algebra VI: Lie 2-algebras". Theory and Applications of Categories. 12: 492–528. Archived from the original on 2018-07-20. Retrieved 2019-05-03.
- 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