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Quivers supporting twisted Calabi-Yau algebras. (English) Zbl 1492.16008

Authors’ abstract: We consider graded twisted Calabi-Yau algebras of dimension 3 which are derivation-quotient algebras of the form \(A=\Bbbk Q/I\), where \(Q\) is a quiver and \(I\) is an ideal of relations coming from taking partial derivatives of a twisted superpotential on \(Q\). We define the type \((M,P,d)\) of such an algebra \(A\), where \(M\) is the incidence matrix of the quiver, \(P\) is the permutation matrix giving the action of the Nakayama automorphism of \(A\) on the vertices of the quiver, and \(d\) is the degree of the superpotential. We study the question of what possible types can occur under the additional assumption that \(A\) has polynomial growth. In particular, we are able to give a nearly complete answer to this question when \(Q\) has at most 3 vertices.

MSC:

16E65 Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.)
16P90 Growth rate, Gelfand-Kirillov dimension
16S38 Rings arising from noncommutative algebraic geometry
16W50 Graded rings and modules (associative rings and algebras)
16T05 Hopf algebras and their applications
16S40 Smash products of general Hopf actions

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References:

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