Some semi-direct products with free algebras of symmetric invariants

TL;DR

This paper investigates the structure of invariants in semi-direct product Lie algebras formed from a reductive Lie algebra and its representations, identifying cases where the invariants form a free algebra, especially for the defining representation of sl_n.

Contribution

It characterizes semi-direct products with free algebra of invariants, particularly those related to the defining representation of sl_n, expanding understanding of symmetry invariants in these Lie algebras.

Findings

Identifies conditions under which the algebra of invariants is free.

Describes all semi-direct products related to the defining representation of sl_n with free invariants.

Provides structural results on the algebra of invariants for these semi-direct products.

Abstract

Let $\mathfrak g$ be a complex reductive Lie algebra and $V$ the underling vector space of a finite-dimensional representation of $\mathfrak g$. Then one can consider a new Lie algebra $\mathfrak q=\mathfrak g{\ltimes} V$, which is a semi-direct product of $\mathfrak g$ and an Abelian ideal $V$. We outline several results on the algebra $\mathbb C[\mathfrak q^*]^{\mathfrak q}$ of symmetries invariants of $\mathfrak q$ and describe all semi-direct products related to the defining representation of $\mathfrak{sl}_n$ with $\mathbb C[\mathfrak q^*]^{\mathfrak q}$ being a free algebra.