Trees, Amalgams and Calogero-Moser Spaces

Details Trees, Amalgams and Calogero-Moser Spaces Trees, Amalgams and Calogero-Moser Spaces

2010-10-25

arxiv.org/abs/1010.5189v1

Mathematics

Quantum Algebra

12 pages, 2 figures

Scientific paper

We describe the structure of the automorphism groups of algebras Morita equivalent to the first Weyl algebra $ A_1 $. 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 r\^ole in our approach is played by a transitive action of the automorphism group of the free algebra $ \c < x, y > $ on the Calogero-Moser varieties $ \CC_n $ defined in \cite{BW}. Our results generalize well-known theorems of Dixmier and Makar-Limanov on automorphisms of $ A_1 $, answering an old question of Stafford (see \cite{St}). Finally, we propose a natural extension of the Dixmier Conjecture for $ A_1 $ to the class of Morita equivalent algebras.

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