Pro-finite p-adic Lie algebras

Abstract

Let p be a prime number. A finite nilpotent Lie ring of characteristic a power of p is called finite- p. A pro- p Lie ring is an inverse limit of finite- p Lie rings. Pro- p Lie rings play a role in Lie theory similar to that played by pro- p groups in group theory. Every pro- p Lie ring admits the structure of a Lie algebra over the p-adic integers; furthermore, every p-adic Lie algebra of finite rank as a p-adic module has an open pro- p subalgebra. We make a detailed study of pro- p Lie rings in terms of various properties, including their topology, Prüfer rank, subring growth, and p-adic module structure. In particular, we prove the equivalence of the following conditions for a finitely generated pro- p Lie ring L: L has finite Prüfer rank; L is isomorphic to a closed subring of gl ( V ) for some p-adic module V of finite rank; and, for sufficiently large n, the Lie F p -subalgebra W n = 〈 e 12 , t e 22 〉 ⊆ gl 2 ( F p [ t ] / 〈 t n 〉 ) is not an open section of L. By reducing to the pro- p Lie ring case, we also prove that all Engelian pro-finite Lie rings are locally nilpotent. This is a Lie theoretic analogue of Zelmanov's theorem which states that every periodic pro- p group is locally finite.