The cohomology of the \(\mathfrak {spl}(2,1)\) superalgebra and reducibility of its indecomposable constituents (Q1078649)

In the paper one of the classical simple Lie superalgebras is studied. Its underlying Lie algebra is \(\mathfrak{sl}(2)\times\mathfrak{gl}(1)\) which has many physical applications. The irreducible representations of \(\mathfrak{spl}(2,1)\) were studied by \textit{M. Scheunert}, \textit{W. Nahm} and \textit{V. Rittenberg} [J. Math. Phys. 18, 155--162 (1977; Zbl 0354.17005)]. In this paper the cohomology of the irreducible representations of the \(\mathfrak{spl}(2,1)\) superalgebra is computed. It turns out that \(H^ n_{\varphi}(\mathfrak{spl}(2,1), W)=0\) if \(n>0\) for all the typical irreducible representations \(W\), and for the trivial representation. For the nontypical irreducible representations \(H^ n(\mathfrak{spl}(2,1), W)=0\) for \(q>n/2\) where \(\varphi: \mathfrak{spl}(2,1)\to \mathrm{End}\;W\). This information is applied in studying the reducibility of the indecomposable representations.