Bounds on the order of generation of SO(n,R) by one-parameter subgroups
Let G be a compact Lie group and \(X_ 1,...,X_ k\) a set of generators of its Lie algebra \({\mathfrak g}\). Then any element of G can be written as a finite product of exponentials exp \(t_ jX_ j\), \(j\in \{1,...,k\}\). The author addresses the question how many factors of this kind one needs. For \(G=SO(n)\) he constructs various sets of generators via a successive use of the Cartan decomposition of G when viewed as the group of isometries of a compact Riemannian symmetric space. Then he gives uniform bounds for the number of factors needed. In the case SO(3) he shows that any two elements of \({\mathfrak g}\) which are not collinear can be used as generators and calculates the minimal number of factors in terms of the angle between the two infinitesimal rotations (i.e. their axes).
scientific article; zbMATH DE number 4202612
scientific article