Cohomology of truncated coinduced representations of Lie algebras of positive characteristic

A. S. Dzhumadil'daev

Sbornik. Mathematics, Tome 66 (1990) no. 2, pp. 461-473

Résumé

The author proves that for any $n$-dimensional Lie algebra of characteristic $p>0$ and any $k$, $0\leqslant k\leqslant n$, there exists a finite-dimensional module with nontrivial $k$-cohomology; the nontrivial cocycles of such modules become trivial under some finite-dimensional extension. He also obtains a criterion for the Lie algebra to be nilpotent in terms of irreducible modules with nontrivial cohomology. The proof of these facts is based on a generalization of Shapiro's lemma. The truncated induced and coinduced representations are shown to be the same thing. Bibliography: 22 titles.