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Indice et decomposition de Cartan d'une algebre de Lie semi-simple reelle | | Anne Moreau
; | | Date: |
10 Jun 2005 | | Subject: | Representation Theory MSC-class: 22E46; 22E15; 17B20; 20G05 | math.RT | | Affiliation: | IMJ | | Abstract: | The Iwasawa decomposition $mathfrak{g}=mathfrak{k}\_0 oplus hat{mathfrak{a}}\_0 oplus mathfrak{n}\_0$ of the real semisimple Lie algebra $mathfrak{g}\_0$ comes from its Cartan decomposition $mathfrak{g}\_0=mathfrak{k}\_0 oplus mathfrak{p}\_0$. Then we get $mathfrak{g}\_0=mathfrak{k}\_0 oplus mathfrak{b}\_0$ where $mathfrak{b}\_0=hat{mathfrak{a}}\_0 oplus mathfrak{n}\_0$. In this note, we establish an explicite formula for the index, ${
m ind} mathfrak{b}$, of $mathfrak{b}$, where $mathfrak{b}$ is the complexification of $mathfrak{b}\_0$. More precisely, we show the following result : $${
m ind} mathfrak{b} = {
m rg mathfrak{g} - {
m rg mathfrak{k},$$ where $mathfrak{g}$ and $mathfrak{k}$ are respectively the complexifications of $mathfrak{g}\_0$ and $mathfrak{k}\_0$. In particular, this answers positively a question by Raïs in cite{Rais} : is the index additive for the following decomposition : $mathfrak{g}\_0=mathfrak{k}\_0 oplus mathfrak{b}\_0$ ? In the proof, we use the Kostant construction and the Cayley transforms. We also give a characterization of the semisimple real Lie algebra $mathfrak{g}\_0$ whose subalgebra $mathfrak{b}\_0$ has a stable form. | | Source: | arXiv, math.RT/0506206 | | Services: | Forum | Review | PDF | Favorites | |
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