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

Latex, 16 pages

References in corpus (2)

• On characteristic equations, trace identities and Casimir operators of simple Lie algebras
• Fermionic realisations of simple Lie algebras

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