Semi-direct products of Lie algebras and covariants

JOURNAL OF ALGEBRA

卷:490

Article

Panyushev, Dmitri I.¹  Yakimova, Oksana S.² 

[1] RAS, Inst Informat Transmiss Problems, Bolshoi Karetnyiper 19, Moscow 127051, Russia

[2] Friedrich Schiller Univ Jena, Inst Math, D-07737 Jena, Germany

关键词: Index of Lie algebra;    Coadjoint representation;    Symmetric invariants;   

DOI  :  10.1016/j.jalgebra.2017.06.036

来源: Elsevier

【 摘 要 】

Let Q be a connected algebraic group with Lie algebra q. Symmetric invariants of q, i.e., the Q-invariants in the symmetric algebra S(q) of q, is a first approximation to the understanding of the coadjoint action (Q : q*) and coadjoint Q-orbits. In this article, we study a class of non-reductive Lie algebras, where the description of the symmetric invariants is possible and the coadjoint representation has a number of nice invariant-theoretic properties. If G is a semisimple group with Lie algebra g and V is G-module, then we define q to be the semi-direct product of g and V. Then we are interested in the case, where the generic isotropy group for the G-action on V is reductive and commutative. It turns out that in this case symmetric invariants of q can be constructed via certain G-equivariant maps from g to V (covariants). (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

Free