Commutators of small elements in compact semisimple groups and Lie algebras (Q2827846)

The openness at the identity element of the commutator map of compact real semisimple Lie algebras and compact semsimple Lie groups is proved, namely, if \(G\) is a real compact semisimple Lie group, then {\parindent=0.6cm\begin{itemize}\item[--] the commutator map \(\mathrm{comm}_{\mathrm g}: \mathrm g\times \mathrm g\ni (x, y)\to [x,y]\in \mathrm g\) is open at \((0,0)\). \item[--] the commutator map \(\mathrm{Comm}_G: G\times G\ni (X,Y)\to XYX^{-1}Y^{-1}\in G\) is open at \((id,id)\) and the result is extended to arbitrary connected semisimple (Lie or algebraic) compact groups. NEWLINENEWLINE\end{itemize}} A good account of background information of the study of the commutator functions is given.