Hochschild homology and cohomology of down-up algebras

TL;DR

This paper provides explicit calculations of Hochschild and cyclic homology and cohomology for a family of down-up algebras, revealing their algebraic invariants and structures.

Contribution

It offers the first detailed, explicit computations of Hochschild and cyclic (co)homology for 3-Calabi-Yau down-up algebras using Koszul resolutions.

Findings

Explicit Hochschild homology and cohomology formulas

Identification of cyclic homology structure

Application of Koszul bimodule resolution techniques

Abstract

We present a detailed computation of the cyclic and the Hochschild homology and cohomology of generic and 3-Calabi-Yau homogeneous down-up algebras. This family was defined by Benkart and Roby in their study of differential posets. Our calculations are completely explicit, by making use of the Koszul bimodule resolution and arguments similar to those used by the second and third authors for Yang-Mills algebras.