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Hochschild homology and cohomology of generalized Weyl algebras: the quantum case. (English. French summary) Zbl 1294.16007
The Hochschild homology and cohomology groups are determined for generalized Weyl algebras $$A$$ based on a polynomial algebra $$k[h]$$ and an automorphism $$\sigma_q$$ of $$k[h]$$ of quantum type. Here $$A = k[h](\sigma_q,a)$$ as in [V. V. Bavula, St. Petersbg. Math. J. 4, No. 1, 71-92 (1992); translation from Algebra Anal. 4, No. 1, 75-97 (1992; Zbl 0807.16027)], where $$k$$ is a field of characteristic zero, $$q\in k\setminus\{0,1\}$$, $$\sigma_q$$ is the $$k$$-algebra automorphism of $$k[h]$$ sending $$h\mapsto qh$$, and $$a$$ is a polynomial in $$k[h]$$ of degree at least $$2$$. The authors compute all the Hochschild homology and cohomology groups of $$A$$. This complements results of M. A. Farinati and the first two authors [Ann. Inst. Fourier 53, No. 2, 465-488 (2003; Zbl 1100.16008)], who computed these groups for $$k[h](\sigma,a)$$ when $$\sigma$$ is an automorphism of translation type ($$h\mapsto h-h_0$$).
The results are highly dependent on whether $$q$$ is a root of unity or not. For example, letting $$c$$ denote the greatest common divisor of $$a$$ and its derivative $$a'$$, one gets $$HH_2(A)=k^{\deg(c)}$$ if $$q$$ is not a root of unity, while $$HH_2(A)=k[h]/(c)\oplus\bigoplus_{r\in e\mathbb Z} k[h^e]$$ if $$q$$ is a primitive $$e$$-th root of unity.

##### MSC:
 16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.) 16S32 Rings of differential operators (associative algebraic aspects) 16S36 Ordinary and skew polynomial rings and semigroup rings 16T20 Ring-theoretic aspects of quantum groups 17B37 Quantum groups (quantized enveloping algebras) and related deformations 16E10 Homological dimension in associative algebras
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##### References:
 [1] Avramov, Luchezar L.; Iyengar, Srikanth, Gaps in Hochschild cohomology imply smoothness for commutative algebras, Math. Res. Lett., 12, 5-6, 789-804, (2005) · Zbl 1101.13018 [2] Avramov, Luchezar L.; Vigué-Poirrier, Micheline, Hochschild homology criteria for smoothness, Internat. Math. Res. Notices, 1, 17-25, (1992) · Zbl 0755.13006 [3] BACH, A Hochschild homology criterium for the smoothness of an algebra, Comment. Math. Helv., 69, 2, 163-168, (1994) · Zbl 0824.13009 [4] Bavula, V. V., Generalized Weyl algebras and their representations, Algebra i Analiz, 4, 1, 75-97, (1992) · Zbl 0807.16027 [5] Bavula, Vladimir, Representation theory of algebras (Cocoyoc, 1994), 18, Global dimension of generalized Weyl algebras, 81-107, (1996), Amer. Math. Soc., Providence, RI · Zbl 0857.16025 [6] Bergh, Petter Andreas; Erdmann, Karin, Homology and cohomology of quantum complete intersections, Algebra Number Theory, 2, 5, 501-522, (2008) · Zbl 1205.16011 [7] Bergh, Petter Andreas; Madsen, Dag, Hochschild homology and global dimension, Bull. Lond. Math. Soc., 41, 3, 473-482, (2009) · Zbl 1207.16006 [8] Buchweitz, Ragnar-Olaf; Green, Edward L.; Madsen, Dag; Solberg, Øyvind, Finite Hochschild cohomology without finite global dimension, Math. Res. Lett., 12, 5-6, 805-816, (2005) · Zbl 1138.16003 [9] Farinati, M. A.; Solotar, A.; Suárez-Álvarez, M., Hochschild homology and cohomology of generalized Weyl algebras, Ann. Inst. Fourier (Grenoble), 53, 2, 465-488, (2003) · Zbl 1100.16008 [10] Han, Yang, Hochschild (co)homology dimension, J. London Math. Soc. (2), 73, 3, 657-668, (2006) · Zbl 1139.16010 [11] Happel, Dieter, Séminaire d’Algèbre Paul Dubreil et Marie-Paul Malliavin, 39ème Année (Paris, 1987/1988), 1404, Hochschild cohomology of finite-dimensional algebras, 108-126, (1989), Springer, Berlin · Zbl 0688.16033 [12] Hochschild, G.; Kostant, Bertram; Rosenberg, Alex, Differential forms on regular affine algebras, Trans. Amer. Math. Soc., 102, 383-408, (1962) · Zbl 0102.27701 [13] Richard, Lionel; Solotar, Andrea, Isomorphisms between quantum generalized Weyl algebras, J. Algebra Appl., 5, 3, 271-285, (2006) · Zbl 1102.16025 [14] Rodicio, Antonio G., Smooth algebras and vanishing of Hochschild homology, Comment. Math. Helv., 65, 3, 474-477, (1990) · Zbl 0726.13008 [15] Rodicio, Antonio G., Commutative augmented algebras with two vanishing homology modules, Adv. Math., 111, 1, 162-165, (1995) · Zbl 0830.13011 [16] Smith, S. P., A class of algebras similar to the enveloping algebra of $${\rm sl}(2),$$ Trans. Amer. Math. Soc., 322, 1, 285-314, (1990) · Zbl 0732.16019 [17] Solotar, Andrea; Vigué-Poirrier, Micheline, Two classes of algebras with infinite Hochschild homology, Proc. Amer. Math. Soc., 138, 3, 861-869, (2010) · Zbl 1227.16011
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