dorsal/arxiv
View SchemaCohomology of Lie superalgebras and of their generalizations
| Authors | M. Scheunert, R. B. Zhang |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9701037 |
| URL | https://arxiv.org/abs/q-alg/9701037 |
| DOI | 10.1063/1.532508 |
| Journal | J.Math.Phys. 39 (1998) 5024-5061 |
Abstract
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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"abstract": "The cohomology groups of Lie superalgebras and, more generally, of color Lie\nalgebras, are introduced and investigated. The main emphasis is on the case\nwhere the module of coefficients is non-trivial. Two general propositions are\nproved, which help to calculate the cohomology groups. Several examples are\nincluded to show the peculiarities of the super case. For L = sl(1|2), the\ncohomology groups H^1(L,V) and H^2(L,V), with V a finite-dimensional simple\ngraded L-module, are determined, and the result is used to show that\nH^2(L,U(L)) (with U(L) the enveloping algebra of L) is trivial. This implies\nthat the superalgebra U(L) does not admit of any non-trivial formal\ndeformations (in the sense of Gerstenhaber). Garland\u0027s theory of universal\ncentral extensions of Lie algebras is generalized to the case of color Lie\nalgebras.",
"arxiv_id": "q-alg/9701037",
"authors": [
"M. Scheunert",
"R. B. Zhang"
],
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],
"doi": "10.1063/1.532508",
"journal_ref": "J.Math.Phys. 39 (1998) 5024-5061",
"title": "Cohomology of Lie superalgebras and of their generalizations",
"url": "https://arxiv.org/abs/q-alg/9701037"
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