Cohomology and deformations of semidirect sums of Lie algebras
type: Item
instance of: scholarly article
title: Cohomology and deformations of semidirect sums of Lie algebras
zbMATH Open document ID: 0978.17012
author: A. Dzhumadil'daev
published in: Doklady Mathematics
publication date: 2001-02-13T00:00:00Z
review text: Let \(L\) be a Lie algebra over a field of characteristic \(p\geq 0\) and \(M\) be a module over \(L\). The cohomology of the semidirect sum \(L+M\) is considered. It is announced that \[ H^*(L+M,L+M)\cong H^*(L,M)\oplus \overline{H^*}(L,C^*(M,L)) \] where \(\overline{H^*}(L,C^*(M,L))\) is the kernel of the natural map \(H^*(L,C^*(M,L))\rightarrow H^*(L,C^*(M,M)).\) Some corollaries for \(H^i(L+M,L+M)\), \(i=1,2\) are formulated. In particular, it is mentioned that if \(M=L^*\) is the coadjoint module, then \(H^1(L,L\wedge L)\) is contained in \(H^2(L+L^*,L+L^*)\). The results of calculation of \(H^1(L,L\wedge L)\) for a Zassenhaus algebra \(W_1(m)\) are given. The corresponding local deformations and their extensions to the second degree are written.
Mathematics Subject Classification ID: 17B56
Mathematics Subject Classification ID: 17B50
zbMATH DE Number: 1564462
zbMATH Keywords: cohomology of Lie algebras
zbMATH Keywords: deformations of Lie algebras
zbMATH Keywords: semidirect sums of Lie algebras
reviewed by: Michael Kuznetsov
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1564462