On the multi-Koszul property for connected algebras

TL;DR

This paper introduces the multi-Koszul algebra concept, generalizing Koszul algebras for graded connected algebras, and explores its properties, examples, and homological computations, including Hochschild cohomology and A-infinity structures.

Contribution

It defines the multi-Koszul property for graded algebras, extends existing notions, and provides new examples and homological tools for analysis.

Findings

Includes super Yang-Mills algebras as examples

Provides an explicit A-infinity algebra structure of the Yoneda algebra

Shows multi-Koszul algebras with finite relations are K_2 algebras

Abstract

In this article we introduce the notion of multi-Koszul algebra for the case of a locally finite dimensional nonnegatively graded connected algebra, as a generalization of the notion of (generalized) Koszul algebras defined by R. Berger for homogeneous algebras, which were in turn an extension of Koszul algebras introduced by S. Priddy. It also extends and generalizes the definition recently introduced by the author and A. Rey. In order to simplify the exposition we consider the minimal graded projective resolution of the algebra A as a bimodule, which may be used to compute the corresponding Hochschild (co)homology groups. This new definition includes several new interesting examples, e.g. the super Yang-Mills algebras introduced by M. Movshev and A. Schwarz, which are not generalized Koszul or even multi-Koszul for the previous definition given by the author and Rey. On the other hand, we provide an equivalent description of the new definition in terms of the Tor (or Ext) groups, and we show that several of the typical homological computations performed for the generalized Koszul algebras are also possible in this more general setting. In particular, we give an explicit description of the A_infinity-algebra structure of the Yoneda algebra of a multi-Koszul algebra. We also show that a finitely generated multi-Koszul algebra with a finite dimensional space of relations is a K_2 algebra in the sense of T. Cassidy and B. Shelton.