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Superpotentials and higher order derivations. (English) Zbl 1219.16016
By a superpotential the authors mean an element \(\omega\) (subject to several conditions) in the path algebra of a quiver. The corresponding derivation-quotient algebra \(D(\omega,k)\) is the path algebra of the quiver modulo the ideal generated by the derivations of \(\omega\) with respect to the paths of length \(k\).
The authors generalize results of M. Dubois-Violette [J. Algebra 317, No. 1, 198-225 (2007; Zbl 1141.17010)] on the one-vertex case to the case when the quiver has several vertices (the base field is replaced by a semisimple algebra). They give a construction of \(D(\omega,k)\) compatible with Morita equivalence, and show that many interesting algebras arise this way, including McKay correspondence algebras for \(\text{GL}_n\) for all \(n\), and \(4\)-dimensional Sklyanin algebras. It is shown that \(N\)-Koszul twisted Calabi-Yau algebras are equivalent to algebras of the form \(A=D(\omega,k)\) such that the asociated complex yields a bimodule resolution of \(A\). As an application a representation theoretic computation of the moduli space of Sklyanin algebras of dimension \(4\) is given.

MSC:
16G20 Representations of quivers and partially ordered sets
16W50 Graded rings and modules (associative rings and algebras)
16S37 Quadratic and Koszul algebras
Software:
GAP
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References:
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