Optimally defined Racah-Casimir operators for su(n) and their eigenvalues for various classes of representations

TL;DR

This paper establishes a canonical method to define Racah-Casimir operators for su(n), derives a general formula for their eigenvalues across representations, and illustrates these results with explicit examples.

Contribution

It introduces a canonical construction of Racah-Casimir operators from Lie algebra cohomology and provides a general eigenvalue formula for any representation of su(n).

Findings

Derived a general eigenvalue formula for Racah-Casimir operators.

Connected eigenvalues to generalized Dynkin indices.

Presented explicit eigenvalue results for fundamental and adjoint representations.

Abstract

This paper deals with the striking fact that there is an essentially canonical path from the $i$-th Lie algebra cohomology cocycle, $i=1,2,... l$, of a simple compact Lie algebra $\g$ of rank $l$ to the definition of its primitive Casimir operators $C^{(i)}$ of order $m_i$. Thus one obtains a complete set of Racah-Casimir operators $C^{(i)}$ for each $\g$ and nothing else. The paper then goes on to develop a general formula for the eigenvalue $c^{(i)}$ of each $C^{(i)}$ valid for any representation of $\g$, and thereby to relate $c^{(i)}$ to a suitably defined generalised Dynkin index. The form of the formula for $c^{(i)}$ for $su(n)$ is known sufficiently explicitly to make clear some interesting and important features. For the purposes of illustration, detailed results are displayed for some classes of representation of $su(n)$, including all the fundamental ones and the adjoint representation.