Journal of Lie Theory Volume 18 (2008) 413–432 c  2008 Heldermann Verlag Non-abelian Cohomology and Extensions of Lie Algebras N. Inassaridze, E. Khmaladze, and M. Ladra ∗ Communicated by K. Strambach Abstract. We introduce the second non-abelian cohomology of Lie algebras with coefficients in crossed modules using the generalised notion of Lie algebra of derivations, and establish a relationship of this cohomology with Lie algebra extensions by crossed modules. We extend the seven-term exact non-abelian cohomology sequence of Guin to a nine-term exact sequence. Mathematics Subject Index 2000: 17B40, 17B56, 18G10, 18G50. Keywords and phrases: Derivations, crossed modules, non-abelian cohomol- ogy,extensions of Lie algebras. 1. Introduction From the 1960s, many authors have attempted to answer the question of what we should mean by non-abelian cohomology of various algebraic structures (see [6, 7, 18, 19]). A convincing answer for groups and Lie algebras was given by Guin [9, 10] in the 1980s. More recently, in the case of groups, H. Inassaridze [13, 14, 15] has demonstrated how Guin’s definition can be naturally extended to higher dimensions. His non-abelian cohomology theory differs from that of Serre [19] and from the setting of various papers on non-abelian cohomology of groups [4, 5, 6]. In [17], using the non-abelian tensor product of Lie algebras of Ellis [8] and its non-abelian left derived functors, we introduced and investigated the non- abelian homology of Lie algebras, generalising the classical Chevalley-Eilenberg homology and Guin’s low dimensional non-abelian homology of Lie algebras [10]. * The first and second authors were supported by the Spanish Ministry of Education and Science grants SAB2004-0211 and SB2004-0144, respectively, and INTAS grant 03-51-3251. The third author was supported by MEC, Grant MTM2006-15338-C02-01 (European FEDER support included) and by Xunta de Galicia, Grant PGIDITI06PXIB371128PR. The authors were also supported by INTAS grant 06-1000017-8609. ISSN 0949–5932 / $2.50 c  Heldermann Verlag