Levi decomposition (Lie group)

This article deals with the Levi decomposition of Lie groups, for the Levi decomposition of Lie algebras see Levi's theorem (Lie algebra) .

In the mathematical field of the theory of Lie groups , the Levi decomposition provides a decomposition of Lie groups as the semi-direct product of a solvable and a reductive Lie group. It results from the Levi decomposition of Lie algebras and is usually used to reduce the study of general Lie groups to the study of semi-simple Lie groups .

It is named after Eugenio Elia Levi .

Levi decomposition and Levi subgroup

Let it be a simply connected Lie group and its radical , i. H. a maximal closed, solvable normal divisor. Then there is a self-contained, simply connected subgroup with , so that the figure ${\displaystyle G}$ ${\displaystyle R}$ ${\displaystyle S\subset G}$ ${\displaystyle R\cap S=\left\{e\right\}}$

${\displaystyle (r,s)\to rs,r\in R,s\in S}$

is a diffeomorphism of the manifold on . ${\displaystyle R\times S}$ ${\displaystyle G}$

The dismantling as a semi-direct product

${\displaystyle G=R\rtimes S}$

is called Levi decomposition (also Levi – Mal'tsev decomposition or Chevalley decomposition ), the subgroup is called Levi subgroup . The Levi decomposition is not clearly determined. ${\displaystyle S}$

In general, unless it is simply contiguous, the Levi subgroup need not be a self-contained subgroup. However, it is always closed when a (not necessarily simply connected) linear group (i.e. a closed subgroup of ) or more generally an algebraic group defined above or above . ${\displaystyle G}$ ${\displaystyle H}$ ${\displaystyle GL(n,\mathbb {C} )}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$

properties

• The Levi subgroup is a reductive group . ${\displaystyle S}$
• A subgroup is the Levi subgroup of a Levi decomposition if and only if it is a maximally reductive subgroup.
• Each reductive subgroup of is conjugated to a subgroup of . ${\displaystyle G}$ ${\displaystyle S}$
• In particular, all Levi subgroups are conjugated to one another.

Mostow's theorem

The set of Mostow says that it for every connected algebraic group over a body of characteristic is a Levi decomposition. For algebraic groups over fields of the characteristic this is generally not correct. ${\displaystyle 0}$ ${\displaystyle p>0}$

Levi-Mostow dismantling of grids

Let it be a simply connected Lie group whose Levi subgroup has no compact factor. Then in every lattice there is a subgroup of finite index that can be continuously deformed into a lattice that is a semi-direct product of the lattice with a lattice . ${\displaystyle G}$ ${\displaystyle S}$ ${\displaystyle \Gamma \subset G}$ ${\displaystyle \Gamma ^{*}\subset \Gamma }$ ${\displaystyle \Gamma ^{\prime }}$ ${\displaystyle \Gamma ^{\prime }=(\Gamma \cap R)\rtimes (\Gamma ^{\prime }\cap S)}$ ${\displaystyle \Gamma \cap R\subset R}$ ${\displaystyle \Gamma ^{\prime }\cap S\subset S}$

literature

• EE Levi: Sulla struttura dei gruppi finiti e continui . In: Atti. Accad. Sci. Torino Cl. Sci. F sharp. Mat. Nature. , 40, 1905, pp. 551-565 (Italian).
• AI Mal'tsev: On the representation of an algebra as a direct sum of the radical and a semi-simple subalgebra . In: Dokl. Akad. Nauk SSSR , 36, 1942, pp. 2, 42–45 (Russian)
• N. Jacobson: Lie algebras. Republication of the 1962 original. Dover Publications, New York 1979, ISBN 0-486-63832-4 .
• AA Kirillov: Elements of the theory of representations. Translated from the Russian by Edwin Hewitt. Basic Teachings of Mathematical Sciences, Volume 220.Springer-Verlag, Berlin / New York 1976.
• MA Naĭmark, AI Štern: Theory of group representations. Translated from the Russian by Elizabeth Hewitt. Translation edited by Edwin Hewitt. Basic teaching of the mathematical sciences 246. Springer-Verlag, New York 1982, ISBN 0-387-90602-9 .

Web links

• Levi-Mal'tsev decomposition . Encyclopedia of Mathematics

Individual evidence

1. ^ Theorem 2.3 in: EB Vinberg, VV Gorbatsevich, OV Shvartsman: Discrete subgroups of Lie groups. In: Encyclopaedia Math. Sci. , 21, Springer, Berlin 2000. Lie groups and Lie algebras, II, pp. 1–123, 217–223,