1997/01/31 by M. Scheunert, R. B. Zhang · 113 citations
Mathematics · #Advanced Topics in Algebra #Algebra over a field #Algebraic structures and combinatorial models #Cohomology #Graded Lie algebra #Group cohomology #Homotopy and Cohomology in Algebraic Topology #Lie algebra #Lie superalgebra #Simple (philosophy) #Simple Lie group #math.QA #q-alg
paper · pdf · doi:10.1063/1.532508
published in Journal of Mathematical Physics 39(9), 5024-5061 (American Institute of Physics) · 50 pages, Latex, no figures. In the revised version the proof of Lemma 5.1 is greatly simplified, some references are added, and a pertinent result on sl(m|1) is announced. To appear in the Journal of Mathematical Physics
arxiv created 1998/04/30 · openalex publication_date 1998/09/01 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
The cohomology groups of Lie superalgebras and, more generally, of ε Lie algebras, are introduced and investigated. The main emphasis is on the case where the module of coefficients is nontrivial. 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 H1(L,V) and H2(L,V), with V a finite-dimensional simple graded L-module, are determined, and the result is used to show that H2(L,U(L)) [with U(L) the enveloping algebra of L] is trivial. This implies that the superalgebra U(L) does not admit any nontrivial formal deformations (in the sense of Gerstenhaber). Garland’s theory of universal central extensions of Lie algebras is generalized to the case of ε Lie algebras.