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Mat. Zametki, 2017, Volume 101, Issue 5, Pages 723–732 (Mi mz11219)  

On the Second Cohomology of an Algebraic Group and of Its Lie Algebra in a Positive Characteristic

Sh. Sh. Ibraevab

a Bolashak University, Kyzylorda
b Institute of Mathematics and Mathematical Modeling, Ministry of Education and Science, Republic of Kazakhstan

Abstract: Necessary and sufficient isomorphism conditions for the second cohomology group of an algebraic group with an irreducible root system over an algebraically closed field of characteristic $p\ge 3h-3$, where $h$ stands for the Coxeter number, and the corresponding second cohomology group of its Lie algebra with coefficients in simple modules are obtained, and also some nontrivial examples of isomorphisms of the second cohomology groups of simple modules are found. In particular, it follows from the results obtained here that, among the simple algebraic groups $\mathrm{SL}_2(k)$, $\mathrm{SL}_3(k)$, $\mathrm{SL}_4(k)$, $\mathrm{Sp}_4(k)$, and $G_2$, nontrivial isomorphisms of this kind exist for $\mathrm{SL}_4(k)$ and $G_2$ only. For $\mathrm{SL}_4(k)$, there are two simple modules with nontrivial second cohomology and, for $G_2$, there is one module of this kind. All nontrivial examples of second cohomology obtained here are one-dimensional.

Keywords: algebraic group, Lie algebra of an algebraic group, simple module, second cohomology group.

Funding Agency Grant Number
Ministry of Education and Science of the Republic of Kazakhstan 0828/ГФ4
This work was supported by the Ministry of Education and Science of the Republic of Kazakhstan under grant 0828/GF4: “Algebras close to Lie ones: cohomologies, identities, and deformations.”


DOI: https://doi.org/10.4213/mzm11219

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English version:
Mathematical Notes, 2017, 101:5, 841–849

Bibliographic databases:

UDC: 512.815.6
Received: 04.05.2016

Citation: Sh. Sh. Ibraev, “On the Second Cohomology of an Algebraic Group and of Its Lie Algebra in a Positive Characteristic”, Mat. Zametki, 101:5 (2017), 723–732; Math. Notes, 101:5 (2017), 841–849

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