On the homology of Lie algebras like $\mathfrak{gl}(\infty, R)$

Abstract

We revisit a recent paper of Fialowski and Iohara. They compute the homology of the Lie algebra gl(∞, R) for R an associative unital algebra over a field of characteristic zero. We explain how to obtain essentially the same results by a completely different method. This note is inspired by the recent paper [FI17] of Fialowski and Iohara. They compute the Lie algebra homology of the infinite matrix algebra gl(∞, R). We shall recall below how this Lie algebra is defined. Their paper takes up a thread of research initiated by Feigin and Tsygan, and generalizes one of the main results of [FT83].While gl(∞, R) naturally acts on Laurent polynomials R[t, t −1 ], in this note we consider a very closely related variant, which acts in an analogous fashion on formal Laurent series R((t)). This leads to a topologically completed variant of the same Lie algebra, call it gl top (∞, R). However, this little change of perspective is only made for convenience, it is really not the main point of this text. Instead, our focus is on computing the homology of gl top (∞, R) in a completely different fashion than the methods used by Fialowski and Iohara.Nonetheless, a posteriori it will turn out that gl(∞, R) and gl top (∞, R) have the same homology.