The representations of solvable Lie algebras with a triangular decomposition
Let \(L\) be a Lie algebra over an algebraically closed field \(F\) of characteristic \(p>0\). If \(L=T\oplus N\) where \(T\) is an \(ad\)-torus and \(N\) a nilpotent subalgebra of \(L\) with \([T,N]\subset N\), then \(L\) is called a solvable Lie algebra with a triangular decomposition (SLATD) (this terminology seems to the reviewer somewhat misleading). The author gives a sufficient condition for a SLATD to be generalized restricted (with respect to a basis \(B\) of \(L\)) in the sense of [\textit{B. Shu}, J. Algebra 194, 157-177 (1997; Zbl 0885.17014)]. For such a generalized restricted SLATD \(L\) and a ``generalized character'' \(S\in \text{Map}(B,F)\) (cf. [B. Shu, ibid.]) such that \(S(B\cap N)=0\), let \(u(S)\) be the \(S\)-reduced enveloping algebra of \(L\). The author determines the irreducible and the indecomposable modules of \(u(S)\). The blocks of \(u(S)\) are described, in particular, the number and the dimensions of the blocks are computed.
scientific article; zbMATH DE number 1671599