I will start the talk with the classical "Cayley transform" for the special orthogonal group
SO(n) constructed by Arthur Cayley in 1846. A connected linear algebraic group G over C is
called a *Cayley group* if it admits a *Cayley map*, that is, a Gequivariant birational
isomorphism between the group variety G and its Lie algebra Lie(G). For example, SO(n) is a
Cayley group. A linear algebraic group G is called *stably Cayley* if G x S is Cayley for some
torus S. I will consider semisimple algebraic groups, in particular, simple algebraic groups. I
will describe classification of Cayley simple groups and of stably Cayley semisimple groups.
(Based on joint works with Boris Kunyavskii and others.)
