Iwasawa decomposition

The Iwasawa decomposition of semi- simple Lie groups generalizes the fact that every square matrix can be represented in a unique way as the product of an orthogonal matrix and an upper triangular matrix. It is named after Kenkichi Iwasawa (1949), who introduced it for real semi-simple Lie groups .

Special case: matrices

A special case is the unambiguous representation of each element of the special linear group as a product of three elements. ${\displaystyle {\textrm {SL}}(n,\mathbb {R} )}$

Let be the special orthogonal group , the set of diagonal matrices with positive diagonal entries, the product of which is, and the set of triangular matrices with all ones on their diagonals. Then there exist for each     uniquely determined     such that . (Compare QR decomposition .) ${\displaystyle K}$ ${\displaystyle {\textrm {SO}}(n,\mathbb {R} )}$ ${\displaystyle A}$ ${\displaystyle 1}$ ${\displaystyle N}$ ${\displaystyle g\in {\textrm {SL}}(n,\mathbb {R} )}$ ${\displaystyle k\in K,a\in A,n\in N}$ ${\displaystyle g=kan}$

General case

Be a semi-simple Lie group . Then there is a decomposition ${\displaystyle G}$

${\displaystyle G=KAN}$

with a compact subgroup , an Abelian subgroup, and a nilpotent subgroup , so that each element is uniquely a product ${\displaystyle K}$ ${\displaystyle A}$ ${\displaystyle N}$ ${\displaystyle g\in G}$

${\displaystyle g=kan}$

with can be disassembled. ${\displaystyle k\in K,a\in A,n\in N}$

The decomposition is not clearly determined. Any decomposition with the above properties is called an Iwasawa decomposition. ${\displaystyle G=KAN}$

The method is named after its developer Iwasawa Kenkichi .

literature

• Lexicon of Mathematics , Spektrum-Akademischer Verlag, 2004, ISBN 3-8274-1159-9
• James Humphreys: Arithmetic groups , Springer-Verlag, 1980, ISBN 0-387-09972-7