On commutators in real semisimple Lie groups
Does every element of a simple group be a commutator ? For finite groups this question was raised by \textit{O. Ore} [Proc. Am. Math. Soc. 2, 307-314 (1951; Zbl 0043.024)], and the answer is still unknown even in this case. For connected complex semisimple Lie groups the answer is positive [\textit{S. Pasiencier} and \textit{H.-C. Wang}, ibid. 13 (1962), 907-913 (1963; Zbl 0112.025)]. The paper under review deals with the ''real'' case. The results are the following: 1) let \(\epsilon =\exp (2\pi i/n)\), \(n=p+q\), \(p\geq q\geq 1\), then \(\epsilon\) is not a commutator in SU(p,q); 2) if a maximal compact subgroup of a connected semisimple Lie group G is semisimple then every \(g\in G\) is a product of two commutators; 3) every element of a special linear group over quaternions is a commutator; 4) the same is true for Sp(p,q) and some other real groups.
scientific article; zbMATH DE number 3975249