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Rigidity of Graded Regular Algebras
| Content Provider | Semantic Scholar |
|---|---|
| Author | Kuzmanovich, James |
| Copyright Year | 2008 |
| 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 An(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. 0. Introduction The invariant theory of k[x1, · · · , xn] is a rich subject whose study has motivated many developments in commutative algebra and algebraic geometry. One important result is the Shephard-Todd-Chevalley Theorem [Theorem 1.1] that gives necessary and sufficient conditions for the fixed subring k[x1, · · · , xn] under a finite subgroup G of GLn(k) to be a polynomial ring. The study of the invariant theory of noncommutative algebras is not well understood, and it is reasonable to begin with the study of finite groups acting on rings that are seen as generalizations of polynomial rings. We will show that in contrast to the commutative case, a noncommutative regular algebra A is often rigid, meaning that A is not isomorphic to any fixed subring A under a nontrivial group of automorphisms G of A. A typical result is the AlevPolo rigidity theorem that shows that both the universal enveloping algebra of a semisimple Lie algebra and the Weyl algebras An(k) are rigid algebras. Theorem 0.1 (Alev-Polo rigidity theorem [AP]). (a) Let g and g′ be two semisimple Lie algebras. Let G be a finite group of algebra automorphisms of U(g) such that U(g) ∼= U(g′). Then G is trivial and g ∼= g′. (b) If G is a finite group of algebra automorphisms of An(k), then the fixed subring An(k) is isomorphic to An(k) only when G is trivial. The main goal of this paper is to investigate a similar question for graded algebras. As one example, in Section 6 we prove the following graded version of the Alev-Polo rigidity theorem. Let H(g) denote the homogenization of the universal enveloping algebra of a finite dimensional Lie algebra g (the definition is given in Section 6). Received by the editors November 6, 2006. 2000 Mathematics Subject Classification. Primary 16E10, 16W30, 20J05. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://arxiv.org/pdf/0706.0662v1.pdf |
| Alternate Webpage(s) | http://www.ams.org/journals/tran/2008-360-12/S0002-9947-08-04571-6/S0002-9947-08-04571-6.pdf |
| Language | English |
| Access Restriction | Open |
| Subject Keyword | Class Emoticon Gareth Rees (software developer) Generalization (Psychology) Kleene algebra Linear algebra MV-algebra Mathematics Subject Classification Muscle Rigidity Polynomial ring Regular expression Ring device Subgroup A Nepoviruses |
| Content Type | Text |
| Resource Type | Article |
