A Cohomological Characterization of Leibniz Central Extensions of Lie Algebras.pdf

a r X i v : m a t h / 0 6 0 5 3 9 9 v 5 [ m a t h .Q A ] 2 8 O c t 2 0 0 6 A COHOMOLOGICAL CHARACTERIZATION OF LEIBNIZ CENTRAL EXTENSIONS OF LIE ALGEBRAS NAIHONG HU?, YUFENG PEI, AND DONG LIU Abstract. Mainly motivated by Pirashvili’s spectral sequences on a Leibniz algebra, a cohomological characterization of Leibniz central extensions of Lie algebras is given based on Corollary 3.3 and Theorem 3.5. In particular, as applications, we obtain the cohomological version of Gao’s main Theorem in [9] for Kac-Moody algebras and answer a question in [16]. 1. Introduction Leibniz algebras introduced by Loday ([17]) are a non-antisymmetric general- ization of Lie algebras. There is a (co)homology theory for these algebraic objects whose properties are similar to those of the classical Chevalley-Eilenberg cohomol- ogy theory for Lie algebras. Since a Lie algebra is a Leibniz algebra, it is interesting to study Leibniz (co)homology of Lie algebras which may provide new invariants for Lie algebras. Lodder ([19]) obtained the Godbillon-Vey invariants for foliations by computing Leibniz cohomology of certain Lie algebras and mentioned how a Leibniz algebra arises naturally from vertex (operator) algebras. Recently, some interrela- tions with manifolds were investigated, which could lead to possible applications of Leibniz (co)homology in geometry (see [11, 19], etc.). Central extensions play a central role in the theory of Lie algebras (see [1, 2, 3, 6, 7, 8, 13, 14, 15, 20, 22], etc.). Viewing a Lie algebra as a Leibniz algebra, it is natural to determine its Leibniz central extensions, and compare the differences between Leibniz and Lie central extensions. Loday-Pirashvili ([18]) have shown that the Virasoro algebra is a universal central extension of the Witt algebra in the category of Leibniz algebras as well. It is well-known that any Kac-Moody Lie algebra g(A) is centrally closed ([8]). However, its universal central extension in the category of Leibniz algebras is