Homological properties of $3$-dimensional DG Sklyanin algebras

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 $(a,b,c)\in {\mathbb{P}}_k^2-\mathfrak{D}$ and $$\mathfrak{D}=\{(1,0,0),(0,1,0),(0,0,1)\}\bigsqcup \{(a,b,c)|a^3=b^3=c^3\}.$$ 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.