2015/02/25 by Izuru Mori, Mori, Izuru, S. Paul Smith +1
Mathematics · #16E65 #16S37 #16S38 #16S80 #16W50 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.1502.07403
openalex publication_date 2015/02/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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(\sfw), the "Jacobian algebra" of some \sfw ∈ V⊗ 3. This paper classifies the \sfw∈ V⊗ 3 such that J(\sfw) is 3-Calabi-Yau. The classification depends on how \sfw transforms under the action of the symmetric group S3 on V⊗ 3 and on the nature of the subscheme \\sfw=0\ ⊆ ℙ2 where \sfw denotes the image of \sfw in the symmetric algebra SV. Surprisingly, as \sfw ranges over V⊗ 3-\0\, only nine isomorphism classes of algebras appear as non-3-Calabi-Yau J(\sfw)'s.