Pro-finite \(p\)-adic Lie algebras. (English) Zbl 1181.17009

Let \(p\) be a prime number, \(\mathbb Z_p\) the ring of \(p\)-adic integers. A Lie ring \(L\) is called finite-\(p\) if it is finite nilpotent Lie \(\mathbb Z_p\)-algebra. A pro-\(p\) Lie ring \(L\) is the inverse limit of an inverse system of finite-\(p\) Lie rings. For example, as it is shown in the paper if \(L\) is a Lie \(\mathbb Z_p\)-algebra of finite rank then \(pL\) is a pro-\(p\) Lie ring. The goal of the paper declared by authors is to study pro-\(p\) Lie rings in the direction analogous to recent studies of pro-\(p\) groups and, thus, to develop understanding of groups as well as Lie rings.
The first part of the paper is devoted to the study of various properties of pro-\(p\) Lie rings concerning their topology, Prüfer rank, subring growth, \(p\)-adic module structure. According to results by Serre for finitely generated pro-\(p\) groups, Hartley, Segal for finitely generated pro-soluble groups, Segal and Nikolov for finitely generated pro-finite groups the topology of such groups is uniquely determined by their algebraic structure. In the second part of the paper it is proved that the topology of finitely generated pro-nilpotent Lie ring is uniquely determined by its algebraic structure. In the third part of the paper the Kurosh problem for pro-finite Lie rings is considered. It is proved that any Engelian pro-finite Lie ring is locally nilpotent. The proof is based on the techniques developed by Wilson and Zelmanov to prove that all periodic pro-finite groups are locally finite and that all Engelian pro-finite groups are locally nilpotent. The fundamental role in the proof plays Zelmanov’s theorem on a finitely generated modular Lie algebra with ad-nilpotent commutators in generators, satisfying non-trivial polynomial identity. At the end of the paper a series of open problems is formulated.


17B60 Lie (super)algebras associated with other structures (associative, Jordan, etc.)
17B50 Modular Lie (super)algebras
20E18 Limits, profinite groups
20F40 Associated Lie structures for groups
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