3rd homotopy group of a compact Simple Lie Group - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T23:55:35Z http://mathoverflow.net/feeds/question/90124 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/90124/3rd-homotopy-group-of-a-compact-simple-lie-group 3rd homotopy group of a compact Simple Lie Group plus111163103100909682983posts 2012-03-03T16:02:41Z 2012-03-03T23:20:14Z <p>Suppose $G$ is a compact simple Lie group with Lie algebra $\mathfrak g$. Then we know that $\pi_3(G)=Z$. Now suppose that $H_\alpha$ is a co-root vector in correspondence with a root $\alpha$. So it means that there are $X_\alpha$ and $Y_\alpha$ such that $span${$H_\alpha, X_\alpha, Y_\alpha$} is a sub-Lie algebra of $\mathfrak g$ isomorphic to $\mathfrak{su}(2)$. It induces a map of Lie groups $\phi:SU(2) \to G$. I'm wondering what's the image of this map as an element of $\pi_3(G)$ in terms of $G$.</p> http://mathoverflow.net/questions/90124/3rd-homotopy-group-of-a-compact-simple-lie-group/90154#90154 Answer by Vitali Kapovitch for 3rd homotopy group of a compact Simple Lie Group Vitali Kapovitch 2012-03-03T22:11:05Z 2012-03-03T23:20:14Z <p>This number is called the index of the map $\phi: SU(2)\to G$. It can be defined for any homomorphism $\phi:H\to G$ where $H$ is simple. Algebraically it can be computed as follows. Since $\mathfrak h$ is simple the restriction of the Killing form of $\mathfrak g$ to $\mathfrak h$ is a constant multiple of the Killing form of $\mathfrak h$. That constant is the index of $\phi$. In the specific case you are asking about for a simple root $\alpha$ the index can also be written as $\frac{(\alpha_{max},\alpha_{max})}{(\alpha,\alpha)}$ where $\alpha_{max}$ is the longest simple root of $\mathfrak g$. Note that from the classification of compact simple Lie groups this can only be equal to 1,2 or 3.</p> <p>See <a href="http://www.ams.org/mathscinet/search/publdoc.html?pg1=MR&amp;s1=1266842" rel="nofollow">Onishchik, "Topology of transitive transformation groups"</a>, §3.10 and §17.2 for details on this.</p>