Trees, Amalgams and Calogero-Moser Spaces

TL;DR

This paper explores the automorphism groups of algebras Morita equivalent to the Weyl algebra, providing a geometric group presentation, and extends classical results and conjectures in this area.

Contribution

It offers a geometric description of automorphism groups via amalgamated products and generalizes key theorems of Dixmier and Makar-Limanov.

Findings

Automorphism groups are described as amalgamated products using Bass-Serre theory.

A transitive action of automorphisms on Calogero-Moser varieties is established.

Extends the Dixmier Conjecture to Morita equivalent algebras.

Abstract

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.