Electronic Research Announcements in Mathematical Sciences (ERA-MS)

On subgroups of the Dixmier group and Calogero-Moser spaces

Pages: 12 - 21, January 2011      doi:10.3934/era.2011.18.12

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Yuri Berest - Department of Mathematics, Cornell University, Ithaca, NY 14853-4201, United States (email)
Alimjon Eshmatov - Department of Mathematics, University of Arizona, Tucson, AZ 85721-0089, United States (email)
Farkhod Eshmatov - Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1043, United States (email)

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 rĂ´le 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.

Keywords:  Weyl algebras, automorphism groups, Calogero-Moser spaces, Bass-Serre theory, Dixmier Conjecture.
Mathematics Subject Classification:  Primary: 16S32; Secondary: 14H70, 16W20, 20E08.

Received: August 2010;      Revised: February 2011;      Available Online: March 2011.