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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 \(SV\).

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)
Full Text: DOI arXiv
[1] Artin, M; Schelter, W, Graded algebras of global dimension 3, Adv. Math., 66, 171-216, (1987) · Zbl 0633.16001
[2] Artin, M., Tate, J., Van den Bergh, M.: Some algebras associated to automorphisms of elliptic curves. In: The Grothendieck Festschrift, vol. I, pp. 33-85, Progr. Math., 86, Birkhäuser, Boston, MA, 1990. MR0917738 (88k:16003) · Zbl 0744.14024
[3] Artin, M; Tate, J; Bergh, M, Modules over regular algebras of dimension 3, Invent. Math., 106, 335-388, (1991) · Zbl 0763.14001
[4] Berger, R., Solotar, A.: A criterion for homogeneous potentials to be 3-Calabi-Yau, arXiv:1203.3029 · Zbl 1132.16017
[5] Bocklandt, R, Graded Calabi Yau algebras of dimension 3, J. Pure Appl. Algebra, 212, 14-32, (2008) · Zbl 1132.16017
[6] Fischer, G.: Plane Algebraic Curves, Student Mathematical Library, 15. American Mathematical Society, Providence, RI (2003). MR1836037 (2002g:14042) · Zbl 1336.16011
[7] Hulek, K.: Elementary Algebraic Geometry. Student Mathematical Library, 20. American Mathematical Society, Providence, RI (2003). MR1799530 (2001k:14001) · Zbl 0763.14001
[8] Popescu-Pampu, P.: Iterating the Hessian: a dynamical system on the moduli space of elliptic curves and dessins d’enfants. In: Noncommutativity and Singularities, Adv. Stud. Pure Math., vol. 55, pp. 83-98. Math. Soc. Japan, Tokyo (2009) · Zbl 1180.14022
[9] Reyes, M; Rogalski, D; Zhang, JJ, Skew Calabi-Yau algebras and homological identities, Adv. Math., 264, 308-354, (2014) · Zbl 1336.16011
[10] Smith, SP, Degenerate 3-dimensional Sklyanin algebras are monomial algebras, J. Algebra, 358, 74-86, (2012) · Zbl 1271.16028
[11] Bergh, M, Existence theorems for dualizing complexes over non-commutative graded and filtered rings, J. Algebra, 195, 662-679, (1997) · Zbl 0894.16020
[12] Yekutieli, A; Zhang, JJ, Homological transcendence degree, Proc. Lond. Math. Soc. (3), 93, 105-137, (2006) · Zbl 1112.16007
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