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The classification of 3-Calabi-Yau algebras with 3 generators and 3 quadratic relations. (English) Zbl 1396.16005
Summary: Let $$k$$ be an algebraically closed field of characteristic not 2 or 3, $$V$$ a 3-dimensional vector space over $$k$$, $$R$$ a 3-dimensional subspace of $$V\otimes V$$, and $$TV/(R)$$ the quotient of the tensor algebra on $$V$$ by the ideal generated by $$R$$. Raf Bocklandt proved that if $$TV/(R)$$ is 3-Calabi-Yau, then it is isomorphic to $$J(\mathsf{w})$$, the “Jacobian algebra” of some $$\mathsf{w}\in V^{\otimes 3}$$. This paper classifies the $$\mathsf{w}\in V^{\otimes 3}$$ such that $$J(\mathsf{w})$$ is 3-Calabi-Yau. The classification depends on how $$\mathsf{w}$$ transforms under the action of the symmetric group $$S_3$$ on $$V^{\otimes 3}$$ and on the nature of the subscheme $$\{\overline{\mathsf{w}}=0\} \subseteq\mathbb P^2$$ where $$\overline{\mathsf{w}}$$ denotes the image of $$\mathsf{w}$$ in the symmetric algebra $$SV$$.

##### MSC:
 16E65 Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.) 16S37 Quadratic and Koszul algebras 16S38 Rings arising from noncommutative algebraic geometry 16S80 Deformations of associative rings 16W50 Graded rings and modules (associative rings and algebras)
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