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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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