Cohomology of generalized restricted Lie algebras

In this note, the generalized restricted Lie algebra, which was introduced by Shu Bin in [J. Algebra 194 (1997) 157–177], is studied. By generalizing the concept of restricted subalgebras and the concept of restricted homomorphism, we show that the second generalized restricted cohomology H L2(L,M) is isomorphic to the equivalence classes of those generalized restricted extension of M by L. For any generalized restricted Lie algebra (L,BL, L) and any generalized restricted L-module M, we show that the sequence is exact.