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)\$.

### On powerful and \$p\$-central restricted Lie algebras

#### Abstract

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)\$.
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Utilizza questo identificativo per citare o creare un link a questo documento: `http://hdl.handle.net/11587/107521`
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