Cohomology of Lie superalgebras and of their generalizations

Details Cohomology of Lie superalgebras and of their generalizations Cohomology of Lie superalgebras and of their generalizations

1997-01-28

arxiv.org/abs/q-alg/9701037v3

J.Math.Phys. 39 (1998) 5024-5061

Mathematics

Quantum Algebra

50 pages, Latex, no figures. In the revised version the proof of Lemma 5.1 is greatly simplified, some references are added, a

Scientific paper

10.1063/1.532508

The cohomology groups of Lie superalgebras and, more generally, of color Lie algebras, are introduced and investigated. The main emphasis is on the case where the module of coefficients is non-trivial. Two general propositions are proved, which help to calculate the cohomology groups. Several examples are included to show the peculiarities of the super case. For L = sl(1|2), the cohomology groups H^1(L,V) and H^2(L,V), with V a finite-dimensional simple graded L-module, are determined, and the result is used to show that H^2(L,U(L)) (with U(L) the enveloping algebra of L) is trivial. This implies that the superalgebra U(L) does not admit of any non-trivial formal deformations (in the sense of Gerstenhaber). Garland's theory of universal central extensions of Lie algebras is generalized to the case of color Lie algebras.

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