Vol. 22, No. 2, 1967

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ISSN: 0030-8730
On the rigidity of semi-direct products of Lie algebras

Roger Wolcott Richardson, Jr.

Vol. 22 (1967), No. 2, 339–344

Roughly speaking, a Lie algebra L is rigid if every Lie algebra near L is isomorphic to L. It is known that L is rigid if the Lie algebra cohomology space H2(L,L) vanishes. In this paper we give an elementary set of necessary and sufficient conditions, independent of Lie algebra cohomology, for the rigidity of a semi-direct product L = S + pW, where ρ is an irreducible representation of a semi-simple Lie algebra S on a vector space W. These conditions lead to a number of new examples of rigid Lie algebras. In particular, we obtain a rigid Lie algebra L with H2(L,L)0.

Mathematical Subject Classification
Primary: 17.30
Received: 24 October 1966
Published: 1 August 1967
Roger Wolcott Richardson, Jr.