The group SU_3 is Cayley

TL;DR

This paper proves that the special unitary group SU_3 over the real numbers is a Cayley group, extending previous results on the Cayley property of SL_3 over algebraically closed fields, and discusses the Cayley property of SL_{3,R}.

Contribution

It establishes that SU_3 over R is a Cayley group, expanding the class of known Cayley groups and extending prior work on SL_3.

Findings

SU_3 is a Cayley group over R

Extension of Popov's 1975 result on SL_3

Discussion on whether SL_{3,R} is Cayley

Abstract

A linear algebraic group G is over a field K is called a Cayley group if it admits a Cayley map, i.e., a G-equivariant K-birational isomorphism between the group variety G and its Lie algebra. We prove that the special unitary group in 3 variables SU_3 is a Cayley group over R, thus extending a result of V.L. Popov, who proved in 1975 that the special linear group in 3 variables SL_3 over an algebraically closed field of characteristic 0 is Cayley. We also discuss the question whether the R-group SL_{3,R} is Cayley.