We consider graded twisted Calabi-Yau algebras of dimension 3 which are derivation-quotient algebras of the form $A = \Bbbk Q/I$, where $Q$ is a quiver and $I$ is an ideal of relations coming from taking partial derivatives of a twisted superpotential on $Q$. We define the type $(M, P, d)$ of such an algebra $A$, where $M$ is the incidence matrix of the quiver, $P$ is the permutation matrix giving the action of the Nakayama automorphism of $A$ on the vertices of the quiver, and $d$ is the degree of the superpotential. We study the question of what possible types can occur under the additional assumption that $A$ has polynomial growth. In particular, we are able to give a nearly complete answer to this question when $Q$ has at most 3 vertices.

中文翻译：

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支持扭曲 Calabi-Yau 代数的箭袋

我们考虑第 3 维的分级扭曲 Calabi-Yau 代数，它们是 $A = \Bbbk Q/I$ 形式的导商代数，其中 $Q$ 是一个颤动，$I$ 是来自取部分的关系的理想$Q$ 上扭曲超势的导数。我们定义了这种代数 $A$ 的类型 $(M, P, d)$，其中 $M$ 是箭袋的关联矩阵，$P$ 是置换矩阵，给出了 $A 的 Nakayama 自同构的作用$ 在箭袋的顶点上，$d$ 是超势的程度。我们研究了在 $A$ 具有多项式增长的附加假设下可能出现哪些类型的问题。特别是，当 $Q$ 最多有 3 个顶点时，我们能够对这个问题给出一个几乎完整的答案。