On powerful and $p$-central restricted Lie algebras

In this note we analyze the analogy between m-potent and p-central restricted Lie algebras and p-groups. For restricted Lie algebras the notion of m-potency has stronger implications than for p-groups. Every finite-dimensional restricted Lie algebra L is isomorphic to $H/H_{[p]}$ for some finite-dimensional p-central restricted Lie algebra H. In particular, for restricted Lie algebras there does not hold an analogue of J.Buckley's theorem. For p odd one can characterize powerful restricted Lie algebras in terms of the cup product map in the same way as for finite p-groups. Moreover, the p-centrality of the finite-dimensional restricted Lie algebra H has a strong implication on the structure of the cohomology ring $H^\bullet(L,F)$.