The split 5-Casimir operator and the structure of $\wedge \mathfrak{ad}^{\otimes 5}$

TL;DR

This paper uses split Casimir operators to decompose the antisymmetric fifth tensor power of the adjoint representation for all Lie algebras, revealing known and new representations.

Contribution

It provides a general decomposition of d^{\u00a05} using split Casimir operators, including a new representation X_5, applicable to all Lie algebras.

Findings

Decomposition includes representations from d^{} and a new representation X_5.

The dimension of X_5 matches previous proposals.

Applicable to all Lie algebras.

Abstract

In the present paper, using the split Casimir operators we have found the decomposition of the antisymmetric part of $\mathfrak{ad}^{\otimes 5}$. This decomposition contains the representations that appeared in the decomposition of $\mathfrak{ad}^{\otimes 4}$ and only one new representation $X_5$. The dimension of this representation has been proposed in [A.J.Macfarlane, H.Pfeiffer, J.Phys.A: Math. Gen. 36 (2003) 2305]. Our decomposition is valid for all Lie algebras.