On the Cayley degree of an algebraic group

TL;DR

This paper introduces the Cayley degree as a new invariant to measure how close a linear algebraic group is to being Cayley, providing bounds for certain groups and extending previous classifications.

Contribution

It defines the Cayley degree for algebraic groups and establishes upper bounds for this invariant for specific classes of groups, advancing understanding of Cayley groups.

Findings

Defined the Cayley degree as a measure of how far a group is from being Cayley.

Proved upper bounds on the Cayley degrees for some algebraic groups.

Extended previous classifications of Cayley groups to include this new invariant.

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.