Hochschild and cyclic homology of central extensions of preprojective algebras of ADE quivers

TL;DR

This paper extends previous work by calculating the entire Hochschild and cyclic homology groups of a central extension of preprojective algebras of ADE quivers, revealing their periodic structure and deformation properties.

Contribution

It generalizes prior results by explicitly computing all Hochschild (co)homology groups and cyclic homology, and describes the universal deformation of the algebra.

Findings

Hochschild and cyclic homology groups are periodic with period 4.

Explicit computation of the first four Hochschild (co)homology groups.

Description of the universal deformation of the algebra.

Abstract

Let A be the central extension of the preprojective algebra of an ADE quiver introduced by P. Etingof and E. Rains in math/0503393. The paper math/0606403 computes the structure of the zeroth Hochschild (co)homology of A. We generalize the results of math/0606403 by calculating the additive structure of all the Hochschild homology and cohomology groups of A and the cyclic homology of A, and to describe the universal deformation of A. Namely, we show that the (co)homology is periodic with period 4, and compute the first four (co)homology groups in each case.