Dixmier groups and Borel subgroups

In this paper, we study a family {Gn}n≥0 of infinite-dimensional (ind-)algebraic groups associated with algebras Morita equivalent to the Weyl algebra A1(C). We give a geometric presentation of these groups in terms of amalgamated products, generalizing classical theorems of Dixmier and Makar-Limanov. Our main result is a classification of Borel subgroups of Gn for all n. We show that the conjugacy classes of non-abelian Borel subgroups of Gn are in bijection with the partitions of n. Furthermore, we prove an infinite-dimensional analogue of the classical theorem of Steinberg [52] that characterizes Borel subgroups in purely group-theoretic terms. Combined together the last two results imply that the Gn are pairwise non-isomorphic as abstract groups. This settles an old question of Stafford [51].