The Second Leibniz Homology Group for Kac-Moody Lie Algebras

Abstract

It is well known that the second homology group of any Kac–Moody Lie algebra and the Virasoro algebra is trivial. This is equivalent to saying that any Kac–Moody Lie algebra (or the Virasoro algebra) is its own universal covering algebra; see, for example, [1] and [4]. Recently, Loday and Pirashvili in [12] proved that the second Leibniz homology group of the Virasoro algebra is also trivial. The Leibniz homology is a noncommutative analogue of Lie algebra homology developed by J-L. Loday (see [11] and [12]). It can be extended for a larger class of algebras called Leibniz algebras. In this paper, we study the second Leibniz homology group HL2(g(A)) of a Kac–Moody Lie algebra g(A). We shall show that HL2(g(A)) = 0 for any non-affine Kac–Moody algebra g(A), but HL2(g(A)) ≠ 0 for any affine Kac–Moody algebra g(A). In the latter case, we use Wilson's idea [15] to carry out HL2(g(A)) explicitly in terms of certain subspaces of Kähler differentials over the Laurent polynomial ring. More precisely, we have the following result. 1991 Mathematics Subject Classification 17B67, 17B55.