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The product in the Hochschild cohomology ring of preprojective algebras of Dynkin quivers. (English) Zbl 1189.16009

Summary: We compute the cup product structure of the preprojective algebras of Dynkin quivers of type \(D\) and \(E\) over a field of characteristic zero. This is a continuation of the work done by P. Etingof, C.-H. Eu [Mosc. Math. J. 7, No. 4, 601-612 (2007; Zbl 1138.16004)] where the additive structure of the Hochschild cohomology (together with its grading) was computed. Together with the results of K. Erdmann, N. Snashall [J. Algebra 205, No. 2, 391-412 (1998; Zbl 0937.16012), ibid. 205, No. 2, 413-434 (1998; Zbl 0937.16013)] (where the \(A\)-case was studied), this yields a complete description of the product in the Hochschild cohomology of \(ADE\) quivers over a field of characteristic zero.

MSC:

16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
16G20 Representations of quivers and partially ordered sets
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References:

[1] Crawley-Boevey, W.; Etingof, P.; Ginzburg, V., Noncommutative geometry and quiver algebras · Zbl 1111.53066
[2] Etingof, P.; Eu, C., Hochschild and cyclic homology of preprojective algebras of ADE quivers · Zbl 1138.16004
[3] Erdmann, K.; Snashall, N., On Hochschild cohomology of preprojective algebras. I, J. Algebra. J. Algebra, J. Algebra, 205, 2, 413-434 (1998), II · Zbl 0937.16013
[4] Eu, C., The calculus structure of the Hochschild homology/cohomology of preprojective algebras of Dynkin quivers · Zbl 1189.16010
[5] Malkin, A.; Ostrik, V.; Vybornov, M., Quiver varieties and Lusztig’s algebra · Zbl 1120.16015
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