Hochschild homology and cohomology of Generalized Weyl algebras: the quantum case  [ Homologie et cohomologie de Hochschild des algèbres de Weyl généralisées : le cas quantique ]
Annales de l'Institut Fourier, Tome 63 (2013) no. 3, pp. 923-956.

Nous déterminons l’homologie et la cohomologie de Hochschild des algèbres de Weyl généralisées de rang un dans le cas quantique sauf dans quelques cas exceptionnels.

We determine the Hochschild homology and cohomology of the generalized Weyl algebras of rank one which are of ‘quantum’ type in all but a few exceptional cases.

DOI : https://doi.org/10.5802/aif.2780
Classification : 16E40,  16E65,  16U80,  16W50,  16W70
Mots clés : algèbre de Weyl généralisée, cohomologie de Hochschild, dimension globale
@article{AIF_2013__63_3_923_0,
     author = {Solotar, Andrea and Su\'arez-Alvarez, Mariano and Vivas, Quimey},
     title = {Hochschild homology and cohomology of Generalized Weyl algebras: the quantum case},
     journal = {Annales de l'Institut Fourier},
     pages = {923--956},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {63},
     number = {3},
     year = {2013},
     doi = {10.5802/aif.2780},
     mrnumber = {3137476},
     zbl = {1294.16007},
     language = {en},
     url = {www.numdam.org/item/AIF_2013__63_3_923_0/}
}
Solotar, Andrea; Suárez-Alvarez, Mariano; Vivas, Quimey. Hochschild homology and cohomology of Generalized Weyl algebras: the quantum case. Annales de l'Institut Fourier, Tome 63 (2013) no. 3, pp. 923-956. doi : 10.5802/aif.2780. http://www.numdam.org/item/AIF_2013__63_3_923_0/

[1] Avramov, Luchezar L.; Iyengar, Srikanth Gaps in Hochschild cohomology imply smoothness for commutative algebras, Math. Res. Lett., Volume 12 (2005) no. 5-6, pp. 789-804 | Article | MR 2189239 | Zbl 1101.13018

[2] Avramov, Luchezar L.; Vigué-Poirrier, Micheline Hochschild homology criteria for smoothness, Internat. Math. Res. Notices (1992) no. 1, pp. 17-25 | Article | MR 1149001 | Zbl 0755.13006

[3] BACH A Hochschild homology criterium for the smoothness of an algebra, Comment. Math. Helv., Volume 69 (1994) no. 2, pp. 163-168 | MR 1282365 | Zbl 0824.13009

[4] Bavula, V. V. Generalized Weyl algebras and their representations, Algebra i Analiz, Volume 4 (1992) no. 1, pp. 75-97 | MR 1171955 | Zbl 0807.16027

[5] Bavula, Vladimir Global dimension of generalized Weyl algebras, Representation theory of algebras (Cocoyoc, 1994) (CMS Conf. Proc.) Volume 18, Amer. Math. Soc., Providence, RI, 1996, pp. 81-107 | MR 1388045 | Zbl 0857.16025

[6] Bergh, Petter Andreas; Erdmann, Karin Homology and cohomology of quantum complete intersections, Algebra Number Theory, Volume 2 (2008) no. 5, pp. 501-522 | Article | MR 2429451 | Zbl 1205.16011

[7] Bergh, Petter Andreas; Madsen, Dag Hochschild homology and global dimension, Bull. Lond. Math. Soc., Volume 41 (2009) no. 3, pp. 473-482 | Article | MR 2506831 | Zbl 1207.16006

[8] Buchweitz, Ragnar-Olaf; Green, Edward L.; Madsen, Dag; Solberg, Øyvind Finite Hochschild cohomology without finite global dimension, Math. Res. Lett., Volume 12 (2005) no. 5-6, pp. 805-816 | Article | MR 2189240 | Zbl 1138.16003

[9] Farinati, M. A.; Solotar, A.; Suárez-Álvarez, M. Hochschild homology and cohomology of generalized Weyl algebras, Ann. Inst. Fourier (Grenoble), Volume 53 (2003) no. 2, pp. 465-488 http://aif.cedram.org/item?id=AIF_2003__53_2_465_0 | Article | Numdam | MR 1990004 | Zbl 1100.16008

[10] Han, Yang Hochschild (co)homology dimension, J. London Math. Soc. (2), Volume 73 (2006) no. 3, pp. 657-668 | Article | MR 2241972 | Zbl 1139.16010

[11] Happel, Dieter Hochschild cohomology of finite-dimensional algebras, Séminaire d’Algèbre Paul Dubreil et Marie-Paul Malliavin, 39ème Année (Paris, 1987/1988) (Lecture Notes in Math.) Volume 1404, Springer, Berlin, 1989, pp. 108-126 | MR 1035222 | Zbl 0688.16033

[12] Hochschild, G.; Kostant, Bertram; Rosenberg, Alex Differential forms on regular affine algebras, Trans. Amer. Math. Soc., Volume 102 (1962), pp. 383-408 | Article | MR 142598 | Zbl 0102.27701

[13] Richard, Lionel; Solotar, Andrea Isomorphisms between quantum generalized Weyl algebras, J. Algebra Appl., Volume 5 (2006) no. 3, pp. 271-285 | Article | MR 2235811 | Zbl 1102.16025

[14] Rodicio, Antonio G. Smooth algebras and vanishing of Hochschild homology, Comment. Math. Helv., Volume 65 (1990) no. 3, pp. 474-477 | Article | MR 1069822 | Zbl 0726.13008

[15] Rodicio, Antonio G. Commutative augmented algebras with two vanishing homology modules, Adv. Math., Volume 111 (1995) no. 1, pp. 162-165 | Article | MR 1317386 | Zbl 0830.13011

[16] Smith, S. P. A class of algebras similar to the enveloping algebra of sl (2), Trans. Amer. Math. Soc., Volume 322 (1990) no. 1, pp. 285-314 | Article | MR 972706 | Zbl 0732.16019

[17] Solotar, Andrea; Vigué-Poirrier, Micheline Two classes of algebras with infinite Hochschild homology, Proc. Amer. Math. Soc., Volume 138 (2010) no. 3, pp. 861-869 | Article | MR 2566552 | Zbl 1227.16011