On subgroups of the Dixmier group and Calogero-Moser spaces

Yuri Berest; Alimjon Eshmatov; Farkhod Eshmatov

doi:10.3934/era.2011.18.12

Article Contents

2011, Volume 18: 12-21. Doi: 10.3934/era.2011.18.12

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On subgroups of the Dixmier group and Calogero-Moser spaces

1.
Department of Mathematics, Cornell University, Ithaca, NY 14853-4201
2.
Department of Mathematics, University of Arizona, Tucson, AZ 85721-0089
3.
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1043

Received: July 31, 2010

Revised: January 31, 2011

Published: December 2010

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Abstract

We describe the structure of the automorphism groups of algebras Morita equivalent to the first Weyl algebra $ A_1(k) $. In particular, we give a geometric presentation for these groups in terms of amalgamated products, using the Bass-Serre theory of groups acting on graphs. A key rôle in our approach is played by a transitive action of the automorphism group of the free algebra $ k< x, y>$ on the Calogero-Moser varieties $ \CC_n $ defined in [5]. In the end, we propose a natural extension of the Dixmier Conjecture for $ A_1(k) $ to the class of Morita equivalent algebras.

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Mathematics Subject Classification: Primary: 16S32; Secondary: 14H70, 16W20, 20E08.

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