Invariant Adjoint Tensors of the Classical Groups

TL;DR

This paper extends the description of invariant polynomials to invariant tensors on classical Lie algebras, providing generators for these tensors based on the Chevalley map and Coxeter's work.

Contribution

It offers a new explicit description of invariant tensors for classical Lie algebras, expanding the classical invariant theory beyond polynomials.

Findings

Provides generators for invariant tensors of classical Lie algebras.

Extends Chevalley's and Coxeter's work to tensor invariants.

Focuses on classical Lie algebras only.

Abstract

For $\mathfrak{g}$ a simple Lie algebra and $G$ its adjoint group, the Chevalley map and work of Coxeter gives a concrete description of the algebra of $G$-invariant polynomials on $\mathfrak{g}$ in terms of traces over various representations. Here we provide an extension of this description to $G$-invariant tensors on $\mathfrak{g}$, although restricted to only providing generators and only for the classical Lie algebras.