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Fixed rings of the Weyl algebra \(A_ 1({\mathbb{C}})\). (English) Zbl 0695.16022
In this interesting note, the fixed rings \(A_ 1({\mathbb{C}})^ G\) under the action of a finite group G on the first complex Weyl algebra \(A_ 1({\mathbb{C}})\) are classified. A crucial fact, due to J. Alev [Lect. Notes Math. 1197, 1-9 (1986; Zbl 0589.16027)], is that every finite subgroup G of \(Aut_{{\mathbb{C}}}(A_ 1({\mathbb{C}}))\) is conjugate to a subgroup of the canonical image of SL(2,\({\mathbb{C}})\) in \(Aut_{{\mathbb{C}}}(A_ 1({\mathbb{C}}))\). Since the finite subgroups of SL(2,\({\mathbb{C}})\) are classically known (they are the finite cyclic and the binary dihedral groups, plus three additional groups), this opens the way for an explicit study of the action of G. The main result of the article states that, if G and H are finite groups of automorphisms of \(A_ 1({\mathbb{C}})\), then \(A_ 1({\mathbb{C}})^ G\cong A_ 1({\mathbb{C}})^ H\) if and only if \(G\cong H\). As an application of Quillen’s theorem on filtered rings, the authors further point out that \(K_ i(A_ 1({\mathbb{C}})^ G)\cong K_ i({\mathbb{C}}G)\) holds for all \(i\geq 0\). In view of the aforementioned classification of possible groups G, this leads to an explicit description of \(K_ 0(A_ 1({\mathbb{C}})^ G)\) in each case. Finally, the trace group (or 0-th Hochschild or cyclic homology group) of \(A_ 1({\mathbb{C}})^ G\) is calculated in the cases where G is cyclic or binary dihedral.
Reviewer: M.Lorenz

MSC:
16W20 Automorphisms and endomorphisms
16E20 Grothendieck groups, \(K\)-theory, etc.
16S34 Group rings
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