Stably Cayley semisimple groups.

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 its Lie algebra Lie$(G)$. A prototypical example is the classical "Cayley transform" for the special orthogonal group ${\mathbf{SO}}_n$ defined by Arthur Cayley in 1846. A linear algebraic group $G$ is called stably Cayley if $G\times S$ is Cayley for some split $k$-torus $S$. We classify stably Cayley semisimple groups over an arbitrary field $k$ of characteristic 0.