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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
GAP
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