The adjoint representation inside the exterior algebra of a simple Lie algebra

TL;DR

This paper investigates the structure of the space of invariants related to the adjoint representation within the exterior algebra of a simple Lie algebra, revealing it as a free module over primitive invariants.

Contribution

It proves that this invariant space forms a free module of specific rank over the algebra generated by primitive invariants, excluding the highest degree one.

Findings

The invariant space is a free module of rank twice the Lie algebra's rank.

It is free over the exterior algebra generated by all primitive invariants except the highest degree.

The result clarifies the module structure of invariants related to the adjoint representation.

Abstract

For a simple complex Lie algebra $\mathfrak g$ we study the space of invariants $A=\left( \bigwedge \mathfrak g^*\otimes\mathfrak g^*\right)^{\mathfrak g}$, (which describes the isotypic component of type $\mathfrak g$ in $ \bigwedge \mathfrak g^*$) as a module over the algebra of invariants $\left(\bigwedge \mathfrak g^*\right)^{\mathfrak g}$. As main result we prove that $A$ is a free module, of rank twice the rank of $\mathfrak g$, over the exterior algebra generated by all primitive invariants in $(\bigwedge \mathfrak g^*)^{\mathfrak g}$, with the exception of the one of highest degree.