Hochschild homology and cohomology of Generalized Weyl algebras: the quantum case
[Homologie et cohomologie de Hochschild des algèbres de Weyl généralisées : le cas quantique]
Solotar, Andrea1 ; Suárez-Alvarez, Mariano1 ; Vivas, Quimey1
1 Departamento de Matemática-IMAS Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, Pabellón 1 1428, Buenos Aires, Argentina.
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 : 10.5802/aif.2780
Classification : 16E40, 16E65, 16U80, 16W50, 16W70
Keywords: generalized Weyl algebra, Hochschild cohomology, global dimension
Mot clés : algèbre de Weyl généralisée, cohomologie de Hochschild, dimension globale
Solotar, Andrea 1 ; Suárez-Alvarez, Mariano 1 ; Vivas, Quimey 1

1 Departamento de Matemática-IMAS Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, Pabellón 1 1428, Buenos Aires, Argentina. 
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     title = {Hochschild homology and cohomology of {Generalized} {Weyl} algebras: the quantum case},
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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/articles/10.5802/aif.2780/

[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 | DOI | MR | Zbl

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

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

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

[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 | Zbl

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

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

[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 | DOI | MR | Zbl

[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 | DOI | Numdam | MR | Zbl

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

[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 | Zbl

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

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

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

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

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

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