On subgroups of the Dixmier group and Calogero-Moser spaces
Yuri Berest - Department of Mathematics, Cornell University, Ithaca, NY 14853-4201, 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 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 . 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.
Received: August 2010; Revised: February 2011; Available Online: March 2011.