Cartan projection

In mathematics , the Cartan projection is an aid in the theory of Lie groups and Lie algebras .

definition

Let it be a semi-simple Lie group with Lie algebra and a Cartan subalgebra . For a root system, let the positive Weyl chamber and . ${\displaystyle G}$ ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle {\mathfrak {a}}\subset {\mathfrak {g}}}$ ${\displaystyle \Sigma }$ ${\displaystyle {\mathfrak {a}}^{+}\subset {\mathfrak {a}}}$ ${\displaystyle A^{+}=\exp({\mathfrak {a}}^{+})}$

Then there is a unique maximally compact subgroup with ${\displaystyle K\subset G}$

${\displaystyle G=KA^{+}K}$

and a clear picture

${\displaystyle \mu \colon G\to A^{+}}$,

so that each can be unambiguously decomposed as having (from dependent) . ${\displaystyle g\in G}$ ${\displaystyle g=k_{1}\mu (g)k_{2}}$ ${\displaystyle g}$ ${\displaystyle k_{1},k_{2}\in K}$

The image is called the Cartan projection . It applies . ${\displaystyle \mu \colon G\to A^{+}}$ ${\displaystyle \mu (g)=KgK\cap A^{+}}$

example

Be it

${\displaystyle G=SL(n,\mathbb {R} ),K=SO(n),A^{+}=\left\{\operatorname {diag} (\sigma _{1},\ldots ,\sigma _{n})\in G:\sigma _{1}\geq \ldots \geq \sigma _{n}>0\right\}}$.

Then the Cartan projection is given by

${\displaystyle \mu (g)=\operatorname {diag} (\sigma _{1}(g),\ldots ,\sigma _{n}(g))}$,

where is the -th eigenvalue of . ${\displaystyle \sigma _{i}(g)^{2}}$ ${\displaystyle i}$ ${\displaystyle g^{t}g}$

Jordan projection

Another continuous projection can be defined by the Jordan decomposition ; it overhangs with the Cartan projection ${\displaystyle \lambda \colon G\to {\overline {\mathfrak {a}}}^{+}}$

${\displaystyle \lambda (g)=\lim _{n\to \infty }{\frac {exp^{-1}(\mu (g^{n}))}{n}}}$

together. In the case one obtains the figure ${\displaystyle G=SL(n,\mathbb {R} )}$

${\displaystyle \lambda (g)=(\log \mid \lambda _{1}(g)\mid ,\log \mid \lambda _{2}(g)\mid ,\ldots ,\log \mid \lambda _{n}(g)\mid )}$,

whereby the eigenvalues ​​(possibly with repetitions) are in ascending order. ${\displaystyle \lambda _{1}(g),\ldots ,\lambda _{n}(g)}$

literature

• Helgason, Sigurdur: Differential geometry, Lie groups, and symmetric spaces. Corrected reprint of the 1978 original. Graduate Studies in Mathematics, 34th American Mathematical Society, Providence, RI, 2001. ISBN 0-8218-2848-7 (Chapter 9)
• Benoist, Yves: Actions propres sur les espaces homogènes réductifs. (Chapter 3) pdf

Web links

• Witte Morris, David: Cartan decomposition subgroups

Individual evidence

1. ^ Benoist: Propriétés asymptotiques des groupes linéaires , GAFA 7 (1997), 1-47