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Graded Calabi Yau algebras of dimension 3. (English) Zbl 1132.16017
A Calabi-Yau algebra of dimension \(3\) is an algebra for which the third power of the shift functor on the bounded derived category of finite-dimensional modules is a Serre functor on this category. In the paper under review the author shows that graded Calabi-Yau algebras of dimension \(3\) are path algebras of quivers with relations derived from superpotentials. The author also gives an alternative condition in terms of exactness of a certain bimodule complex. It is further shown that for a given quiver \(Q\) and degree \(d\) the set of those superpotentials of degree \(d\), which give rise to Calabi-Yau quotients of the path algebra of \(Q\), is either empty or contains almost all superpotentials (in the measure theoretic sense). Finally, the author applies Gröbner basis techniques to show how one can determine the possible degrees of “good” superpotentials for some simple quivers.

MSC:
16E65 Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.)
16G20 Representations of quivers and partially ordered sets
18E30 Derived categories, triangulated categories (MSC2010)
18G10 Resolutions; derived functors (category-theoretic aspects)
Software:
GAP
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