☆ 4.1 Article

Shephard-Todd-Chevalley Theorem for Skew Polynomial Rings

ALGEBRAS AND REPRESENTATION THEORY (2010)

Journal

ALGEBRAS AND REPRESENTATION THEORY

Volume 13, Issue 2, Pages 127-158

Publisher

SPRINGER

DOI: 10.1007/s10468-008-9109-2

Keywords

Artin-Schelter regular algebra; Group action; Reflection; Trace; Hilbert series; Fixed subring; Quantum polynomial rings

Categories

Mathematics

Funding

National Science Foundation of USA
Royalty Research Fund of the University of Washington

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Abstract

We prove the following generalization of the classical Shephard-Todd-Chevalley Theorem. Let G be a finite group of graded algebra automorphisms of a skew polynomial ring A := k(pij) [x(1), . . . , x(n)]. Then the fixed subring A(G) has finite global dimension if and only if G is generated by quasi-reflections. In this case the fixed subring A(G) is isomorphic to a skew polynomial ring with possibly different p(ij)'s. Aversion of the theorem is proved also for abelian groups acting on general quantum polynomial rings.

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