The Classification of 3-calabi-yau Algebras with 3 Generators and 3 Quadratic Relations

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 ⊗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(w), the “Jacobian algebra” of some w ∈ V . This paper classifies the w ∈ V ⊗3 such that J(w) is 3-Calabi-Yau. The classification depends on how w transforms under the action of the symmetric group S3 on V ⊗3 and on the nature of the subscheme {w = 0} ⊆ P where w denotes the image of w in the symmetric algebra SV . Surprisingly, as w ranges over V ⊗3 − {0}, only nine isomorphism classes of algebras appear as non-3-Calabi-Yau J(w)’s.