Cohomology of generalized restricted Lie algebras
type: Item
instance of: scholarly article
title: Cohomology of generalized restricted Lie algebras
zbMATH Open document ID: 1070.17006
author: Zhihong Jiang
author: Guangyu Shen
published in: Journal of Algebra
publication date: 2004-09-23T00:00:00Z
review text: According to \textit{Shu Bin} [J. Algebra 194, 157--177 (1997; Zbl 0885.17014)] a generalized restricted Lie algebra is a triple \((L, B_L, \varphi _s)\) where \(L\) is a Lie algebra over a field \(F\) of characteristic \(p>0\), \(B_L=\{x_1,\ldots ,x_n\}\) is a basis of \(L\), \(s=(s_1,\ldots ,s_n)\) is an \(n\)-tuple of natural numbers, and \(\varphi _s:B_L\rightarrow L\) is a map such that \(\operatorname{ad} \varphi _s (x_i)=(\operatorname{ad} x_i)^{p^{s_i}}\). Given \((L, B_L, \varphi _s)\), an associative algebra \(u_{\varphi _s}(L)= U(L)/I(B_L, \varphi _s)\) is defined where \(I(B_L, \varphi _s)\) is an ideal generated by \(\varphi _s(x_i)-x_i^{p^{s_i}}\), \(x_i\in B_L \). An \(u_{\varphi _s}(L)\)-module \(M\) is called a generalized restricted \(L\)-module. The authors investigate the relation between \(H^n_{\varphi _s}(L,M)= \text{Ext}^n_{u_{\varphi _s}(L)}(F,M)\) and the usual cohomology \(H^n(L,M)\) for \(n=1,2\). For \(n=1\) they obtain a result generalizing the classical description of the restricted cohomology given by G. P. Hochschild. For \(n=2\) it is proved that \(H^2_{\varphi _s}(L,M)\) is isomorphic to the set of equivalence classes of generalized restricted extensions of \(M\) by \(L\) where \(M\) is a generalized restricted \(L\)-module such that \( M^{[p]}=0\). It is shown that the sequence \[ \begin{multlined} 0\rightarrow H^1_{\varphi _s}(L,M)\rightarrow H^1(L,M) \rightarrow \text{hom}_F(L,M^L)\\ \rightarrow H^2_{\varphi _s}(L,M)\rightarrow H^2(L,M) \rightarrow \text{hom}_F(L,H^1(L,M)) \end{multlined} \] is exact.
reviewed by: Michael Kuznetsov
Mathematics Subject Classification ID: 17B50
Mathematics Subject Classification ID: 17B56
zbMATH DE Number: 2102466
zbMATH Keywords: generalized restricted Lie algebra
zbMATH Keywords: extension
zbMATH Keywords: cohomology
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2004.03.017
OpenAlex ID: W2068727973
Wikidata QID: Q115351736
cites work: ON THE COHOMOLOGY OF MODULAR LIE ALGEBRAS
cites work: On the cohomology of restricted lie algebras
cites work: Cohomology of Restricted Lie Algebras
cites work: Q4780571
cites work: The generalized restricted representations of graded Lie algebras of Cartan type
cites work: Q4039778
DOI: 10.1016/J.JALGEBRA.2004.03.017
description: scientific article; zbMATH DE number 2102466