N. Lemire, Z. Reichstein, V. L. Popov, Cayley groups , J. Amer. Math. Soc. 19 (2006), 921-967.

Abstract: The classical Cayley map, X --> (I_n-X)/(I_n+X) is a birational isomorphism between the special orthogonal group SO_n and its Lie algebra so_n, which is SO_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 exists if and only if n <= 3. This answers an old question of Luna.

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