Abstract
We describe the multiplicative structure of the Hochschild cohomology ring H H ∗(Λ) of the generalized preprojective algebra \(\Lambda =\mathbb {B}_{n}\). This is done by giving the structure of the cohomology groups as modules over the center of Λ and by giving a presentation of H H ∗(Λ), as a bigraded algebra, by means of generators and relations.
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Andreu, E.: The Hochschild cohomology ring of preprojective algebras of type Ln. J. Pure Appl. Algebra 217(8), 1447–1475 (2013)
Andreu, E., Saorín, M.: The symmetry, period and Calabi-Yau dimension of finite dimensional mesh algebras. Preprint available at arXiv:1304.0586 (2013)
Bialkowski, J., Erdmann, K., Skowroński, A.: Deformed preprojective algebras of generalized Dynkin type. Trans. Amer. Math. Soc. 359, 2625–2650 (2007)
Buchweitz, R.O.: Maximal Cohen-Macaulay modules and Tate cohomology over Gorenstein rings. Preprint available at https://tspace.library.utoronto.ca/handle/1807/16682 (1986)
Dugas, A.: Resolutions of mesh algebras: periodicity and Calabi-Yau dimensions. Mathematische Zeitschrift 271(3–4), 1151–1184 (2012)
Erdmann, K., Snashall, N.: On Hochschild cohomology of preprojective algebras I. J. Algebra 205, 391–412 (1998)
Erdmann, K., Snashall, N.: On Hochschild cohomology of preprojective algebras II. J. Algebra 205, 413–434 (1998)
Eu, C.: The product in the Hochschild cohomology ring of preprojective algebras of Dynkin quivers. J. Algebra 320, 1477–1530 (2008)
Eu, C., Schedler, T.: Calabi-Yau Frobenius algebras. J. Algebra 321, 774–815 (2009)
Gabriel, P.: The universal cover of a represenation-finite algebra. Proc. Conference on Repr. Algebras, Puebla 1981. Springer LNM 903, 68–105 (1981)
Gerstenhaber, M.: The cohomology structure of an associative ring. Ann. Math. 78(2), 267–288 (1963)
Green, E.L., Solberg, O., Zacharia, D.: Minimal projective resolutions. Trans. Amer. Math. Soc. 353, 2915–2939 (2001)
Maclane, S.: Homology. Die Grund. der Math. Wiss, vol. 114. Academic Press (1963)
Nastasescu, C., Van Oystaeyen, F.: Graded Ring Theory. North Holland (1982)
Tate, J.: The higher dimensional cohomology groups of class field theory. Ann. Math. 56(2), 294–297
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Presented by Raymundo Bautista.
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Andreu Juan, E., Saorín, M. The Hochschild Cohomology Ring of the Generalized Preprojective Algebra \(\mathbb {B}_{n}\) . Algebr Represent Theor 17, 1721–1770 (2014). https://doi.org/10.1007/s10468-014-9468-9
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DOI: https://doi.org/10.1007/s10468-014-9468-9