Semi-direct products involving $Sp_{2n}$ or $Spin_n$ with free algebras of symmetric invariants

TL;DR

This paper classifies semi-direct products involving symplectic or orthogonal Lie algebras and modules, identifying cases where the symmetric invariants form a free algebra, advancing understanding of invariant theory in Lie algebra extensions.

Contribution

It provides a classification of semi-direct products with free symmetric invariant algebras for representations of $Sp_{2n}$ and $Spin_n$, filling a gap in the theory.

Findings

Classified semi-direct products with free symmetric invariants for $Sp_{2n}$ and $Spin_n$.

Identified specific modules leading to free invariant algebras.

Contributed to the broader project of classifying Lie algebra extensions with free invariants.

Abstract

This is a part of an ongoing project, the goal of which is to classify all semi-direct products $\mathfrak s=\mathfrak g{\ltimes} V$ such that $\mathfrak g$ is a simple Lie algebra, $V$ is a $\mathfrak g$-module, and $\mathfrak s$ has a free algebra of symmetric invariants. In this paper, we obtain such a classification for the representations of the orthogonal and symplectic algebras.