Dixmier groups and Borel subgroups

Abstract In this paper, we study a family { G n } n ≥ 0 of infinite-dimensional (ind-)algebraic groups associated with algebras Morita equivalent to the Weyl algebra A 1 ( 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 G n for all n . We show that the conjugacy classes of non-abelian Borel subgroups of G n 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 G n are pairwise non-isomorphic as abstract groups. This settles an old question of Stafford [51] .