Computation of \(H^*(sl(2),M)\) with coefficients in a simple \(sl(2)\)- module (Q1200378)

Cohomologies of irreducible representations of the three-dimensional simple Lie algebra \({\mathfrak sl}(2)\) over the field of characteristic zero are calculated. For finite-dimensional representations the answer to this question is simple and well-known. So, the problem is actual only for infinite-dimensional modules. It is proved that among irreducible weight modules only two Verma modules have nonzero cohomologies. For nonweight modules formulae are given allowing to calculate first and second cohomologies. In particular the triviality of the cohomology groups of Whittaker modules is obtained. Except for Whittaker modules there exists a series of irreducible \({\mathfrak sl}(2)\)-modules having rank 1 module structure over \(K[H]\), where \(\langle H\rangle\) is the Cartan subalgebra. The nontriviality of cohomology groups of these modules is also obtained. Cohomologies of irreducible representations of \({\mathfrak sl}(2)\) in nonzero characteristic cases were calculated by the reviewer [Mat. Sb., Nov. Ser. 119(161), No. 1, 132-149 (1982; Zbl 0504.17006)].