On Restricted Cohomology of Modular Classical Lie Algebras and Their Applications
Ibraev S.S. Kainbaeva L. Seitmuratov A.
May-2 2022MDPI
Mathematics
2022#10Issue 10
In this paper, we study the restricted cohomology of Lie algebras of semisimple and simply connected algebraic groups in positive characteristics with coefficients in simple restricted modules and their applications in studying the connections between these cohomology with the corresponding ordinary cohomology and cohomology of algebraic groups. Let G be a semisimple and simply connected algebraic group G over an algebraically closed field of characteristic p > h, where h is a Coxeter number. Denote the first Frobenius kernel and Lie algebra of G by G1 and g, respectively. First, we calculate the restricted cohomology of g with coefficients in simple modules for two families of restricted simple modules. Since in the restricted region the restricted cohomology of g is equivalent to the corresponding cohomology of G1, we describe them as the cohomology of G1 in terms of the cohomology for G1 with coefficients in dual Weyl modules. Then, we give a necessary and sufficient condition for the isomorphisms Hn(G1, V) ≅ Hn (G, V) and Hn(g, V) ≅ Hn(G, V), and a necessary condition for the isomorphism Hn(g, V) ≅ Hn(G1, V), where V is a simple module with highest restricted weight. Using these results, we obtain all non-trivial isomorphisms between the cohomology of G, G1, and g with coefficients in the considered simple modules.
Algebraic group , Cohomology , Lie algebra , Restricted cohomology , Simple modules
Text of the article Перейти на текст статьи
Department of Physics and Mathematics, Institute of Natural Science, Korkyt Ata Kyzylorda Univesity, Aiteke bie St. 29A, Kyzylorda, 120014, Kazakhstan
Department of Physics and Mathematics
10 лет помогаем публиковать статьи Международный издатель
Книга Публикация научной статьи Волощук 2026 Book Publication of a scientific article 2026