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.
