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Hochschild homology and cohomology of generalized Weyl algebras. (English) Zbl 1100.16008

Summary: We compute Hochschild homology and cohomology of a class of generalized Weyl algebras, introduced by V. V. Bavula [in St. Petersbg. Math. J. 4, No. 1, 71-92 (1993); translation from Algebra Anal. 4, No. 1, 75-97 (1992; Zbl 0807.16027)]. Examples of such algebras are the \(n\)-th Weyl algebras, \({\mathcal U}(\mathfrak{sl}_2)\), primitive quotients of \({\mathcal U}(\mathfrak{sl}_2)\), and subalgebras of invariants of these algebras under finite cyclic groups of automorphisms. We answer a question of V. V. Bavula and D. A. Jordan [Trans. Am. Math. Soc. 353, No. 2, 769-794 (2001; Zbl 0961.16016)] concerning the generators of the group of automorphisms of a generalized Weyl algebra. We also explain previous results on the invariants of Weyl algebras and of primitive quotients.

MSC:

16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
17B35 Universal enveloping (super)algebras
16S36 Ordinary and skew polynomial rings and semigroup rings
17B37 Quantum groups (quantized enveloping algebras) and related deformations
16W20 Automorphisms and endomorphisms
16W22 Actions of groups and semigroups; invariant theory (associative rings and algebras)
16S32 Rings of differential operators (associative algebraic aspects)
16W35 Ring-theoretic aspects of quantum groups (MSC2000)
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References:

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