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On the Cayley degree of an algebraic group

Nicole Lemire·Vladimir L. Popov·Zinovy Reichstein
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Abstract

A connected linear algebraic group G is called a Cayley group if the Lie algebra of G endowed with the adjoint G-action and the group variety of G endowed with the conjugation G-action are birationally G-isomorphic. In particular, the classical Cayley map, X \mapsto (I_n-X)/(I_n+X), between the special orthogonal group SO_n and its Lie algebra so_n, shows that SO_n is a Cayley group. In an earlier paper (see math.AG/0409004) we classified the simple Cayley groups defined over an algebraically closed field of characteristic zero. Here we consider a new numerical invariant of G, the Cayley degree, which "measures" how far G is from being Cayley, and prove upper bounds on Cayley degrees of some groups.

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Nicole Lemire, Vladimir L. Popov, Zinovy Reichstein. 2006-08-18. On the Cayley degree of an algebraic group. https://arxiv.org/abs/math/0608473

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