Cohomology and deformations of semidirect sums of Lie algebras
Let \(L\) be a Lie algebra over a field of characteristic \(p\geq 0\) and \(M\) be a module over \(L\). The cohomology of the semidirect sum \(L+M\) is considered. It is announced that \[ H^*(L+M,L+M)\cong H^*(L,M)\oplus \overline{H^*}(L,C^*(M,L)) \] where \(\overline{H^*}(L,C^*(M,L))\) is the kernel of the natural map \(H^*(L,C^*(M,L))\rightarrow H^*(L,C^*(M,M)).\) Some corollaries for \(H^i(L+M,L+M)\), \(i=1,2\) are formulated. In particular, it is mentioned that if \(M=L^*\) is the coadjoint module, then \(H^1(L,L\wedge L)\) is contained in \(H^2(L+L^*,L+L^*)\). The results of calculation of \(H^1(L,L\wedge L)\) for a Zassenhaus algebra \(W_1(m)\) are given. The corresponding local deformations and their extensions to the second degree are written.
scientific article; zbMATH DE number 1564462