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Register a new userCalabi-Yau properties of $3$-dimensional DG Sklyanin algebras
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In this paper, we compute all possible differential structures of a $3$-dimensional DG Sklyanin algebra $mathcal{A}$, which is a connected cochain DG algebra whose underlying graded algebra $mathcal{A}^{#}$ is a $3$-dimensional Sklyanin algebra $S_{a,b,c}$. We show that there are three major cases depending on the parameters $a,b,c$ of the underlying Sklyanin algebra $S_{a,b,c}$: (1) either $a^2 eq b^2$ or $c eq 0$, then $partial_{mathcal{A}}=0$; (2) $a=-b$ and $c=0$, then the $3$-dimensional DG Sklyanin algebra is actually a DG polynomial algebra; and (3) $a=b$ and $c=0$, then the DG Sklyanin algebra is uniquely determined by a $3times 3$ matrix $M$. It is worthy to point out that case (2) has been systematically studied in cite{MGYC} and case (3) is just the DG algebra $mathcal{A}_{mathcal{O}_{-1}(k^3)}(M)$ in cite{MWZ}. We solve the problem on how to judge whether a given $3$-dimensional DG Sklyanin algebra is Calabi-Yau.
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