$s$-homogeneous algebras via $s$-homogeneous triples

TL;DR

This paper introduces the concept of $s$-homogeneous triples and establishes their equivalence with $s$-homogeneous algebras, providing new insights into their properties, relations, and classifications, especially regarding $s$-Koszul algebras.

Contribution

It develops the theory of $s$-homogeneous triples, links them to $s$-homogeneous algebras, and classifies certain $s$-Koszul algebras with specific relations and Veronese rings.

Findings

Equivalence between $s$-homogeneous algebras and $s$-homogeneous triples.

Classification of $s$-Koszul algebras with one relation.

Identification of $s$-homogeneous algebras with specific Veronese rings.

Abstract

To study $s$-homogeneous algebras, we introduce the category of quivers with $s$-homogeneous corelations and the category of $s$-homogeneous triples. We show that both of these categories are equivalent to the category of $s$-homogeneous algebras. We prove some properties of the elements of $s$-homogeneous triples and give some consequences for $s$-Koszul algebras. Then we discuss the relations between the $s$-Koszulity and the Hilbert series of $s$-homogeneous triples. We give some application of the obtained results to $s$-homogeneous algebras with simple zero component. We describe all $s$-Koszul algebras with one relation recovering the result of Berger and all $s$-Koszul algebras with one dimensional $s$-th component. We show that if the $s$-th Veronese ring of an $s$-homogeneous algebra has two generators, then it has at least two relations. Finally, we classify all $s$-homogeneous algebras with $s$-th Veronese rings ${\bf k}\langle x,y\rangle/(xy,yx)$ and ${\bf k}\langle x,y\rangle/(x^2,y^2)$. In particular, we show that all of these algebras are not $s$-Koszul while their $s$-homogeneous duals are $s$-Koszul.