Some conjectures and results concerning the homology of nilpotent Lie algebras

Abstract

In this paper we consider the Lie algebra homology ofLk=L[circle times operator](C[t]/(tk+1))for L a complex Lie algebra. Our goal is to express the homology of Lk in terms of the homology of L. This problem comes up in previous work by the author on the Macdonald root system conjectures.We present a number of conjectures related to this problem. The simplest of these conjectures asserts thatH(Lk)=H(L)[circle times operator](k+1)when L is either semisimple or a nilpotent upper summand of a semisimple Lie algebra. We give a proof of (*) in the case L = sln(). Lastly we present computational evidence in support of our other conjectures.