Superpotentials and higher order derivations

By Raf Bocklandt, Travis Schedler, M. Wemyss. 2008

138 citations recorded by the source corpus.

Abstract We consider algebras defined from quivers with relations that are k th order derivations of a superpotential, generalizing results of Dubois-Violette to the quiver case. We give a construction compatible with Morita equivalence, and show that many important algebras arise in this way, including McKay correspondence algebras for GL n for all n , and four-dimensional Sklyanin algebras. More generally, we show that any N -Koszul, (twisted) Calabi–Yau algebra must have a (twisted) superpotential, and construct its minimal resolution in terms of derivations of the (twisted) superpotential. This yields an equivalence between N -Koszul twisted Calabi–Yau algebras A and algebras defined by a superpotential ω such that an associated complex is a bimodule resolution of A . Finally, we apply these results to give a description of the moduli space of four-dimensional Sklyanin algebras using the Weil representation of an extension of SL 2 ( Z / 4 ) .

Subfields: Algebra and Number Theory, Geometry and Topology, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/91ddc9edc7e07604964f0148492abf84a3346e91

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Citations in the corpus

Skew Calabi–Yau algebras and homological identities (2013, background): “graded or ﬁltered algebras when rigid dualizing complexes were studied in the late 1990s. The skew CY property was called “rigid Gorenstein” in [BZ, Deﬁnition 4.4], was called “twisted Calabi-Yau” in [BSW] (the word “twisted” was alsoused in the work [DV]), and mentioned in talks by several other pe”
Stable categories of Cohen-Macaulay modules and cluster categories: Dedicated to Ragnar-Olaf Buchweitz on the occasion of his sixtieth birthday (2011, methodology): “mal Cohen-Macaulay R-modules is a (d− 1)-Calabi-Yau triangulated category. (b) [Aus86, Yos90, IT10] We have EndR(S) ≃ S ∗G. (c) [Iya07a] The R-module S is a (d−1)-cluster tilting object in CM(R). (d) [BSW10] S ∗ G is presented by the McKay quiver of G with relations given by higher derivative of a p”
Calabi-Yau algebras and superpotentials (2010, methodology): “t (−) sends left l-modules to right l-modules and vice versa. The challenge is to make the (−)∗-operation compatible with the monoidal structure given by the tensor product over l. It is explained in [2] how to do this. For the rest of this paper we ﬁx a trace Tr : l→kand we let σ= σ′ ⊗σ′′ ∈l⊗k l be”
Representations of Algebras and Related Topics (2011, background): “ived (n + 1)-preprojective algebra Πn+1(Λ) as the DG algebra TΛΘn, where Θn is a coﬁbrant resolution of RHomΛe (Λ,Λe)[n]. Then Corollary 3.21 generalizes when replacing 2 by n. In recent papers [53], [19], [68], [91], generalizations of Ginzburg DG algebras are introduced and studied. Special case n”
Flops and clusters in the homological minimal model programme (2014, background): “there is an MM generator (in fact, CT module) given by the skew group ring := EndR(N ), and further by [3] it can be presented as”