On Cohomology of Simple Modules for Modular Classical Lie Algebras

Sherali S Ibraev,Larissa S Kainbaeva,Saulesh K Menlikozhaeva

doi:10.3390/axioms11020078

Sherali S Ibraev, Larissa S Kainbaeva + Show 1 more

Open Access

https://doi.org/10.3390/axioms11020078

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Journal: Axioms	Publication Date: Feb 16, 2022
Citations: 1	License type: CC BY 4.0

Affiliation: Korkyt Ata Kyzylorda State University

Abstract

In this article, we obtain some cohomology of classical Lie algebras over an algebraically closed field of characteristic p>h, where h is a Coxeter number, with coefficients in simple modules. We assume that these classical Lie algebras are Lie algebras of semisimple and simply connected algebraic groups. To describe the cohomology of simple modules, we will use the properties of the connections between ordinary and restricted cohomology of restricted Lie algebras.

Highlights

The Lie algebra cohomology was first introduced in 1929 by Cartan for a one-dimensional trivial module in order to extend de Rham s cohomological methods for Lie algebras [1]
We have calculated the cohomology of classical Lie algebras over an algebraically closed field of characteristic p > h, where h is a Coxeter number, with coefficients in simple modules whose highest weights belong to alcoves along the walls of the dominant Weyl chambers and close to them alcoves
We were interested in studying the cohomology of simple modules for g, G1, and G in the restricted region, and the connections between them

Summary

Introduction

The Lie algebra cohomology was first introduced in 1929 by Cartan for a one-dimensional trivial module in order to extend de Rham s cohomological methods for Lie algebras [1]. We have calculated the cohomology of classical Lie algebras over an algebraically closed field of characteristic p > h, where h is a Coxeter number, with coefficients in simple modules whose highest weights belong to alcoves along the walls of the dominant Weyl chambers and close to them alcoves. The description of the cohomology of classical Lie algebras over an algebraically closed field of characteristic zero with coefficients in a trivial one-dimensional module is known [34] Theorem 1, given, is devoted to the cohomology of classical Lie algebras with coefficients in simple modules, isomorphic to quotient modules of Weyl modules with a simple radical by the maximal submodules. Theorem 2 describes the cohomology of classical Lie algebras with coefficients in some simple modules, which are isomorphic to quotient modules of Weyl modules with a nonsimple radical by the maximal submodules.

Calculation Algorithm of the Cohomology with Coefficients in Simple Modules

Formulation of Results

Proof of the Results

E8 F4 G2 i wi νi

Discussion