Piecewise-Koszul algebras

TL;DR

This paper introduces the concept of piecewise-Koszul algebras, a new class of quadratic algebras that generalize classical Koszul algebras, providing criteria and structural insights into their properties.

Contribution

It defines piecewise-Koszul algebras, establishes a criterion for their characterization via Yoneda-Ext algebras, and explores their relation to classical and higher Koszul algebras.

Findings

Criteria theorem for piecewise-Koszul algebras

Existence of an $A_{ abla}$-structure on $E(A)$

Relations between Koszul and piecewise-Koszul algebras

Abstract

It is a small step toward the Koszul-type algebras. The piecewise-Koszul algebras are, in general, a new class of quadratic algebras but not the classical Koszul ones, simultaneously they agree with both the classical Koszul and higher Koszul algebras in special cases. We give a criteria theorem for a graded algebra $A$ to be piecewise-Koszul in terms of its Yoneda-Ext algebra $E(A)$, and show an $A_{\infty}$-structure on $E(A)$. Relations between Koszul algebras and piecewise-Koszul algebras are discussed. In particular, our results are related to the third question of Green-Marcos' [6].