Semi-direct products of Lie algebras and covariants

Abstract

The coadjoint representation of a connected algebraic group $Q$ with Lie algebra $\mathfrak{q}$ is a thrilling and fascinating object. Symmetric invariants of $\mathfrak{q}$ (= $\mathfrak{q}$-invariants in the symmetric algebra $S(\mathfrak{q})$) can be considered as a first approximation to the understanding of the coadjoint action $(Q:{\mathfrak{q}}^{\ast })$ and coadjoint 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 $\mathfrak{g}$ and $V$ is $G$-module, then we define $\mathfrak{q}$ to be the semi-direct product of $\mathfrak{g}$ 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 $\mathfrak{q}$ can be constructed via certain $G$-equivariant maps from $\mathfrak{g}$ to $V$ ("covariants").

33 pages

References in corpus (2)

• Some semi-direct products with free algebras of symmetric invariants
• On maximal commutative subalgebras of Poisson algebras associated with involutions of semisimple Lie algebras