A characterization of the differential in semi-infinite cohomology

TL;DR

This paper develops a foundational understanding of semi-infinite cohomology, defining its differential as a universal inner derivation and proving its properties with a concise, rigorous approach.

Contribution

It introduces a new characterization of the differential in semi-infinite cohomology as a universal inner derivation, providing a simplified proof of its square-zero property.

Findings

Differential d is the unique derivation satisfying the Cartan identity.

d is shown to be square-zero with a concise proof.

Semi-infinite cohomology generalizes finite-dimensional Lie algebra cohomology.

Abstract

Semi-infinite cohomology is constructed from scratch as the proper generalization of finite dimensional Lie algebra cohomology. The differential d and other operators are realized as universal inner deri- vations of a completed algebra, which acts on any appropriate semi-infinite complex. In particular, d is shown to be the unique derivation satisfying the "Cartan identity" and certain natural degree conditions. The proof that d is square-zero may well be the shortest (arguably, the only) one in print.