Stably Cayley groups

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 Lemire, Popov and Reichstein,
who classified Cayley and stably Cayley simple groups over an
algebraically closed field of characteristic zero. I will explain
their results and also some new results from my recent preprint with
Lemire, Kunyavskii and Reichstein, where we study Cayley and stably
Cayley reductive groups over an arbitrary field k of characteristic
zero.