Cayley groups

Abstract

The classical Cayley map, X ↦ ( I n − X ) ( I n + X ) − 1 X \mapsto (I_n-X)(I_n+X)^{-1} , is a birational isomorphism between the special orthogonal group SO n _n and its Lie algebra s o n {\mathfrak so}_n , which is SO n _n -equivariant with respect to the conjugating and adjoint actions, respectively. We ask whether or not maps with these properties can be constructed for other algebraic groups. We show that the answer is usually “no", with a few exceptions. In particular, we show that a Cayley map for the group SL n _n exists if and only if n ⩽ 3 n \leqslant 3 , answering an old question of Luna .