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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.

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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