Compatible Iwasawa decomposition for embedding of the orthogonal Lie group - MathOverflow 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 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 Answer by Aakumadula for Compatible Iwasawa decomposition for embedding of the orthogonal Lie group 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>