Classification of simple sl(2)-modules and finite-dimensionality of the module of extensions of simple sl(2)-modules (Q753925)

Let (H,X,Y) be the standard basis of the Lie algebra \({\mathfrak sl}(2)\) such that \([H,X]=2X\), \([H,Y]=-2Y\), \(H=[X,Y]\) and U the universal enveloping algebra of \({\mathfrak sl}(2)\). An \({\mathfrak sl}(2)\)-module V is said to be weighted (generally weighted) if \(V=\oplus V_{\lambda}\) \((V=\oplus V^{\lambda}\) where \(V_{\lambda}=Ker(H-\lambda)\), \(V^{\lambda}=\cup_{n}Ker(H-\lambda)^ n\). The classification of weighted \({\mathfrak sl}(2)\)-modules was presented by \textit{D. Arnal} and \textit{G. Pinczon} [J. Math. Phys. 15, 350-359 (1974; Zbl 0298.17003)]. The author considers non-weighted modules. He introduces a generalization of Whittaker modules and Arnal-Pinczon series. The author proves that dim Ext\({}_{{\mathfrak sl}(2)}(M,N)<\infty\) if M,N are simple \({\mathfrak sl}(2)\)- modules. Also, Ker \(u_ M<\infty\), Coker \(u_ M<\infty\) if \(u\in U\), \(u_ M\neq 0\).