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Journal of the London Mathematical Society 2000 62(2):381-397; doi:10.1112/S0024610700001204
© 2000 by London Mathematical Society
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© The London Mathematical Society

On the Rigidity of Lie Lattices and Just Infinite Powerful Groups

Thomas Weigel

Mathematical Institute, University of Oxford 24–29 St Giles, Oxford OX1 3LB

Received 19 March 1999. Revision received 9 August 1999.

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 L be a semisimple complex Lie algebra of dimension d and let B be a C-basis of L. Let Formula denote the affine variety of all structure constants of C-Lie algebras of dimension d and let c isin L(d) denote the point corresponding to the structure constants of L with respect to B. Then there exists an open neighbourhood in the metric topology U sub L(d) of c isin L(d) such that Lc* is isomorphic to L for all c* isin U, where Lc* denotes the Lie algebra defined by the structure constants c* isin 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.


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