"locally" factoring subgroups of Lie groups - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T22:14:20Z http://mathoverflow.net/feeds/question/61304 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/61304/locally-factoring-subgroups-of-lie-groups "locally" factoring subgroups of Lie groups Starting_Stats 2011-04-11T15:23:03Z 2011-04-11T16:25:02Z <p>I'm not really a math person, and apologize if the question here is too simple. I've ended up with the following type of question for a few Lie groups, but state it for SO(n).</p> <p>I start with a subgroup G of SO(n) generated by rotations in $k$ fixed 2-dimensional planes $P_{1}, \ldots, P_{k}$. If I fix a permutation $\sigma \in S_{k}$, can I generate any element of $G$ that is 'sufficiently close' to the identity by multiplying rotations in the planes $P_{\sigma(1)}, \ldots, P_{\sigma(k)}$ in that order?</p> <p>I know that this is nonsense for discrete groups. I don't care too much about the metric as long as it is 'reasonable' (i.e. any of the $L^{p}$ norms from being in Euclidean space, or Hilbert-Schmidt, are fine; 0-1 metric not so much). If the above factorization is possible, I'd also be interested in knowing if it ever requires going 'very far' from the origin in order to hit elements that are 'very close' to the origin.</p> http://mathoverflow.net/questions/61304/locally-factoring-subgroups-of-lie-groups/61308#61308 Answer by Richard Borcherds for "locally" factoring subgroups of Lie groups Richard Borcherds 2011-04-11T16:25:02Z 2011-04-11T16:25:02Z <p>This is false for G=SO(3). This can be generated by rotations in the two planes orthogonal to the x axis and the y axis, but not every element close to the identity can be written as a product of a rotation in the first plane and a rotation in the second plane. </p>