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

TL;DR

This paper introduces divided power (co)homology for modular Lie superalgebras, corrects existing notions of generators and Cartan matrices, and provides explicit presentations of simple finite-dimensional cases across various characteristics.

Contribution

It develops new notions of divided power (co)homology, corrects existing definitions of Chevalley generators and Cartan matrices, and offers explicit presentations of simple modular Lie superalgebras.

Findings

Introduces divided power (co)homology for modular Lie superalgebras.

Provides presentations of finite-dimensional Lie (super)algebras with Cartan matrices.

Corrects existing notions of Chevalley generators and Cartan matrices.

Abstract

For modular Lie superalgebras, new notions are introduced: Divided power homology and divided power cohomology. For illustration, we give presentations (in terms of analogs of Chevalley generators) of finite dimensional Lie (super)algebras with indecomposable Cartan matrix in characteristic 2 (and in other characteristics for completeness of the picture). We correct the currently available in the literature notions of Chevalley generators and Cartan matrix in the modular and super cases, and an auxiliary notion of the Dynkin diagram. In characteristic 2, the defining relations of simple classical 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 are also given..