most recent 30 from http://mathoverflow.net 2013-05-23T09:32:19Z http://mathoverflow.net/feeds/question/38011 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/38011/compatible-iwasawa-decomposition-for-embedding-of-the-orthogonal-lie-group Eachara Donk 2010-09-07T21:49:15Z 2012-12-22T06:22:00Z

<p>I am looking for an embedding of the orthogonal Lie group O(n,C) into GL(m,C) such that the standard Iwasawa decomposition (also known as the QR-decomposition) for the group GL(m,C) induces an Iwasawa decomposition for the group O(n,C). Recall that the standard Iwasawa decomposition for the general linear group GL(m,C)=U(m)R, with R being the subgroup of upper-diagonal matrices with positive real diagonal entries, and U(m) - the unitary subgroup. </p>

http://mathoverflow.net/questions/38011/compatible-iwasawa-decomposition-for-embedding-of-the-orthogonal-lie-group/115773#115773 Aakumadula 2012-12-08T05:42:18Z 2012-12-08T05:42:18Z

<p>There are general theorems for an arbitrary semi-simple subgroup of $GL_n({\mathbb C})$ which do this. I will work this out for $O(n,{\mathbb C})$ when $n=2m$ is even. </p> <p>Fix the standard inner product with respect to which the unitary group $U(n)$ is defined. View $O(n)=O(2m)$ as the subgroup which fixes the quadratic form </p> <p>$$Q=x_1x_{2m}+x_2x_{2m-1}+\cdots+ x_mx_{m+1}.$$</p> <p>All this is explained in Professor Jim Humphrey's book on Lie algebras.</p> <p>Then the Iwasawa decomposition of $GL_n({\mathbb C})$ restriced to $O(n)$ gives an Iwasawa decomposition on $O(n)$. </p>