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Invariants for automorphisms of certain iterated skew polynomial rings. (English) Zbl 0864.16027

This article studies the action of finite cyclic groups \(C_n\) on a certain special class of skew polynomial algebras \(R\) which includes, among other examples, the ordinary and quantized Weyl algebras in two variables and the enveloping algebra \(U(\text{sl}_2)\). Assuming \(n\) to be nonzero in the base field, the invariant algebra \(S=R^{C_n}\) is shown to be a homomorphic image of a skew polynomial algebra constructed in an analogous fashion as \(R\). Thus information about the representation theory of \(R\), previously accumulated by the first author in a series of articles, can be applied to study finite-dimensional \(S\)-modules. For \(R=U(\text{sl}_2)\), the results imply that \(S\) has \(n^2\) non-isomorphic simple modules in each dimension. This has been independently observed by H. Kraft and L. W. Small [Math. Res. Lett. 1, No. 3, 297-307 (1994; Zbl 0849.16036)] who gave a more broadly based treatment of the connections between the finite-dimensional representations of an algebra and those of its invariant subalgebras under finite group actions.

MSC:

16S36 Ordinary and skew polynomial rings and semigroup rings
16W20 Automorphisms and endomorphisms
17B37 Quantum groups (quantized enveloping algebras) and related deformations
16G30 Representations of orders, lattices, algebras over commutative rings

Citations:

Zbl 0849.16036
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References:

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