Cohomology of generalized restricted Lie algebras
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.
scientific article; zbMATH DE number 2102466