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# Homology, Homotopy and Applications

## Volume 12 (2010)

### Number 1

### Divided power (co)homology. Presentations of simple finite dimensional modular Lie superalgebras with Cartan matrix

Pages: 237 – 278

DOI: http://dx.doi.org/10.4310/HHA.2010.v12.n1.a13

#### Authors

#### Abstract

For modular Lie superalgebras, new notions are introduced: Divided power homology and divided power cohomology. For illustration, we explicitly give presentations (in terms of analogs of Chevalley generators) of finite dimensional Lie (super)algebras with indecomposable Cartan matrix in characteristic 2 (and—in the arXiv version of the paper—in other characteristics for completeness of the picture). In the modular and super cases, we define notions of Chevalley generators and Cartan matrix, and an auxiliary notion of the Dynkin diagram. The relations of simple Lie algebras of the A, D, E types are not only Serre ones. These non-Serre relations are same for Lie superalgebras with the same Cartan matrix and any distribution of parities of the generators. Presentations of simple orthogonal Lie algebras having no Cartan matrix (indigenous for characteristic 2) are also given.

#### Keywords

divided power cohomology; defining relation; modular Lie superalgebra

#### 2010 Mathematics Subject Classification

17B50, 70F25

Published 1 January 2010