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Abstract. We prove a graded version of Alev-Polo's rigidity theorem: the homogenization of the universal enveloping algebra of a semisimple Lie algebra and the Rees ring of the Weyl algebras A n (k) cannot be isomorphic to their fixed subring under any finite group action. We also show the same result for other classes of graded regular algebras including the Sklyanin algebras.

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Abstract. We prove the following generalization of the classical ShephardTodd-Chevalley Theorem. Let G be a finite group of graded algebra automorphisms of a skew polynomial ring A := kp ij [x 1 , · · · , xn]. Then the fixed subring A G has finite global dimension if and only if G is generated by quasireflections. In this case the fixed subring A G is isomorphic a skew polynomial ring with possibly different p ij 's. A version of the theorem is proved also for abelian groups acting on general quantum polynomial rings.

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