# Mathematics Colloquia and Seminars

### Cayley Groups

Algebra & Discrete Mathematics

 Speaker: Nicole Lemire, University of Western Ontario Location: 693 Kerr Start time: Fri, Oct 29 2004, 12:10PM

An algebraic group G is called Cayley if there exists a birational isomorphism between G and its Lie algebra g which is equivariant with respect to the conjugation action of G on itself and the adjoint action of G on g. Cayley was the first to construct such a birational equivalence for SO_n. Luna asked whether or not maps with these properties can be constructed for other algebraic groups. We prove that the answer to Luna's question is usually ''no'' with a few exceptions. In particular, a Cayley map for the group SL_n exists if and only if n \le 3. The negative results are proved by methods of integral representation theory.