Homological properties of $3$-dimensional DG Sklyanin algebras

TL;DR

This paper introduces and studies 3-dimensional DG Sklyanin algebras, analyzing their differential structures and homological properties, including conditions for being Calabi-Yau, Koszul, Gorenstein, and homologically smooth.

Contribution

It systematically investigates the differential and homological properties of 3D DG Sklyanin algebras, establishing conditions for key properties like Calabi-Yau and Koszul.

Findings

Conditions for DG Sklyanin algebras to be Calabi-Yau

Criteria for Koszul and Gorenstein properties

Characterization of homological smoothness

Abstract

In this paper, we introduce the notion of DG Sklyanin algebras, which are connected cochain DG algebras whose underlying graded algebras are Sklyanin algebras. Let $\mathcal{A}$ be a $3$-dimensional DG Sklyanin algebra with $\mathcal{A}^{\#}=S_{a,b,c}$, where $(a,b,c)\in \Bbb{P}_k^2-\mathfrak{D}$ and $$\mathfrak{D}=\{(1,0,0), (0,1,0),(0,0,1)\}\sqcup\{(a,b,c)|a^3=b^3=c^3\}.$$ We systematically study its differential structures and various homological properties. Especially, we figure out the conditions for $\mathcal{A}$ to be Calabi-Yau, Koszul, Gorenstein and homologically smooth, respectively.