On the coadjoint representation of \mathbb Z₂-contractions of reductive Lie algebras

2006/10/16 by Dmitri I. Panyushev, Panyushev, Dmitri I.
Mathematics · #Algebraic Geometry (math.AG) #FOS: Mathematics #Representation Theory (math.RT) #math.AG #math.RT

paper · pdf · doi:10.48550/arxiv.math/0610493

25 pages, 3 tables

arxiv created 2006/10/16 · arxiv updated 2009/12/01

Abstract

We study the coadjoint representation of contractions of reductive Lie algebras associated with symmetric decompositions. Let \frak g=\frak g₀⊕ \frak g₁ be a symmetric decomposition of a reductive Lie algebra \frak g. Then the semi-direct product of \frak g₀ and the \frak g₀-module \frak g₁ is a contraction of \frak g. We conjecture that these contractions have many properties in common with reductive Lie algebras. In particular, it is proved that in many cases the algebra of invariants is polynomial. We also discuss the so-called "codim--2 property" for coadjoint representations and its relationship with the structure of algebra of invariants.

Related

• Semi-direct products of Lie algebras and their invariants
• A remarkable contraction of semisimple Lie algebras
• Contractions of Lie algebras with 2-dimensional generic coadjoint orbits
• Deformations of semi-direct products
• The index of centralizers of elements of reductive Lie algebras