Dixmier groups and Borel subgroups

作者：Berest Yuri^*; Eshmatov Alimjon; Eshmatov Farkhod

来源：Advances in Mathematics, 2016, 286: 387-429.

DOI：10.1016/j.aim.2015.09.012

• Ind-algebraic groups
• Rings of differential operators
• Weyl algebra
• Amalgamated products
• Dixmier Conjecture

In this paper, we study a family {G(n)}n >= 0 of infmitedimensional (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 infinitedimensional 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 GT, are pairwise non-isomorphic as abstract groups. This settles an old question of Stafford [51].

• 出版日期2016-1-2