Stably Cayley groups in characteristic zero

TL;DR

This paper characterizes when reductive algebraic groups over fields of characteristic zero are stably Cayley, providing criteria based on their character lattices and classifying simple cases beyond algebraically closed fields.

Contribution

It offers a new criterion for stably Cayley reductive groups in characteristic zero and classifies stably Cayley simple groups over arbitrary fields.

Findings

Criterion for stably Cayley reductive groups based on character lattices

Classification of stably Cayley simple groups over arbitrary fields

Extension of previous classifications to non-algebraically closed fields

Abstract

A linear algebraic group G over a field k is called a Cayley group if it admits a Cayley map, i.e., a G-equivariant birational isomorphism over k between the group variety G and the Lie algebra Lie(G). A Cayley map can be thought of as a partial algebraic analogue of the exponential map. A prototypical example is the classical "Cayley transform" for the special orthogonal group SO_n defined by Arthur Cayley in 1846. A k-group G is called stably Cayley if the product of G with a split r-dimensional k-torus is Cayley for some r=0,1,2,.... These notions were introduced in 2006 by N. Lemire, V. L. Popov and Z. Reichstein, who classified Cayley and stably Cayley simple groups over an algebraically closed field of characteristic zero. In this paper we study Cayley and stably Cayley reductive groups over an arbitrary field k of characteristic zero. Our main results are a criterion for a reductive k-group G to be stably Cayley, formulated in terms of its character lattice, and the classification of stably Cayley simple (but not necessarily absolutely simple) groups.