Homology of the Lie algebra $\mathfrak{gl}(\infty,R)$

TL;DR

This paper computes the homology of the infinite-dimensional Lie algebra gl(\u221e,R) over a unital associative algebra R, confirming a historical result and connecting to soliton theory.

Contribution

It provides a homology computation for gl(,R), extending previous results and justifying earlier findings in the context of higher-dimensional soliton theory.

Findings

Homology of gl(,R) computed for characteristic zero fields.

Results validate an old theorem of Feigin and Tsygan from 1983.

Special case R= relates to soliton theory applications.

Abstract

In this note we compute the homology of the Lie algebra $\mathfrak{gl}(\infty,R)$ where $R$ is an associative unital $k$-algebra which is used in higher dimensional soliton theory. When $k$ is a field of characteristic $0$, our result justifies an old result of Feigin and Tsygan appeared in 1983. The special case when $R=k$ is the complex number field $\mathbb{C}$ appeared first in soliton theory.