A Converse to the Third Whitehead Lemma

We show that finite-dimensional Lie algebras over a field of characteristic zero such that their high-degree cohomology in any finite-dimensional non-trivial irreducible module vanishes, are, essentially, direct sums of semisimple and nilpotent algebras. The classical First and Second Whitehead Lemmata state that the first, respectively second, cohomology group of a finite-dimensional semisimple Lie algebra over a field of characteristic zero with coefficients in any finite-dimensional module vanishes. This pattern breaks down, however, at the third cohomology: it is well-known that the third cohomology of a simple Lie algebra with coefficients in the trivial one-dimensional module is one-dimensional, the basic cocycle being constructed from the Killing form. So, taking literally, the “Third Whitehead Lemma” does not exist. There is, however, a very similar and not less classical result holding for all higher cohomology groups which could bear such a name: for any finite-dimensional semisimple Lie algebra L and any non-trivial finite-dimensional irreducible L-module V , H(L, V ) = 0 for any n ≥ 3. It is very natural to ask (and, indeed, was asked by Dietrich Burde whom I grateful for an interesting correspondence) whether a converse to these statements holds. A converse to the Second Whitehead Lemma was established in [Z]. The aim of this note, which could be considered as a postscript to [Z], is to observe that a converse to the “Third Whitehead Lemma” readily follows from the results already established in the literature. In what follows, all algebras and modules assumed to be finite-dimensional, and the base field is of characteristic zero. Finite-dimensionality is obviously crucial in most of the places. Some reasonings below are valid in any characteristic, but the case of positive characteristic is trivial modulo existing results, as noted in [Z]. Our notations are standard: for a Lie algebra L and an L-module V , H(L, V ) denotes the nth cohomology of L with coefficients in V , V L denotes the submodule of L-invariant points, Z(L) and Rad(L) denotes the center and the solvable radical of L respectively, V ∗ denotes an L-module adjoint to V . When considered as an L-module, the base field K is understood as the trivial one-dimensional module. Tr(A) denotes the trace of a linear operator A. Theorem. For a Lie algebra L, the following are equivalent: (i) L is the direct sum of a semisimple algebra and a nilpotent algebra. (ii) H(L, V ) = 0 for any n and any non-trivial irreducible L-module V . (iii) H (L, V ) = 0 for any non-trivial irreducible L-module V . (iv) H(L, V ) = 0 for any non-trivial irreducible L-module V . Date: August 1, 2008.