ON THE RIGIDITY OF LIE LATTICES AND JUST INFINITE POWERFUL GROUPS

Abstract

A classical result of M. Gerstenhaber [5] states that finite-dimensional semisimple complex Lie algebras are rigid. One interpretation of this fact is as follows. Let [Lfr ] be a semisimple complex Lie algebra of dimension d and let [Bfr ] be a [Copf ]-basis of [Lfr ]. Let L(d) ⊂ [Copf ]^d·(d2) denote the affine variety of all structure constants of [Copf ]-Lie algebras of dimension d and let c ∈ L(d) denote the point corresponding to the structure constants of [Lfr ] with respect to [Bfr ]. Then there exists an open neighbourhood in the metric topology U ⊂ L(d) of c ∈ L(d) such that [Lfr ]_c* is isomorphic to [Lfr ] for all c* ∈ U, where [Lfr ]_c* denotes the Lie algebra defined by the structure constants c* ∈ L(d). Our aim is to generalize this result to semisimple Lie algebras over discrete valuation domains and to apply these results to powerful pro-p-groups.