(a,b)-Koszul algebras

TL;DR

This paper introduces the concept of (a,b)-Koszul algebras, generalizing N-Koszul algebras, and provides examples, resolutions, and Hochschild homology computations for these new algebraic structures.

Contribution

It defines (a,b)-Koszul algebras, extending the class of N-Koszul algebras, and offers explicit resolutions and homology calculations for them.

Findings

Defined (a,b)-Koszul algebras as a new class

Provided minimal graded projective resolutions

Computed Hochschild homology for examples

Abstract

Let $a$ and $b$ be two integers such that $2\le a<b$. In this article we define the notion of $(a,b)$-Koszul algebra as a generalization of $N$-Koszul algebras. We also exhibit examples and we provide a minimal graded projective resolution of the algebra $A$ considered as an $A$-bimodule, which allows us to compute the Hochschild homology groups for some examples of $(a,b)$-Koszul algebras.