An Explicit Construction of Casimir Operators and Eigenvalues : I

TL;DR

This paper presents a general method for explicitly constructing all Casimir operators of a Lie algebra, including detailed calculations for 4th and 5th order operators of $A_N$ algebras, using polynomial-based coefficients.

Contribution

It introduces a systematic approach to construct complete sets of Casimir operators with explicit polynomial coefficients, applicable to any rank N Lie algebra.

Findings

Explicit formulas for 4th and 5th order Casimir operators of $A_N$ Lie algebras.

A polynomial-based framework for describing symmetric coefficients of Casimir operators.

The number of polynomial clusters is independent of the algebra's rank, enabling generalization.

Abstract

We give a general method to construct a complete set of linearly independent Casimir operators of a Lie algebra with rank N. For a Casimir operator of degree p, this will be provided by an explicit calculation of its symmetric coefficients $ g^{A_1,A_2,.. A_p}$. It is seen that these coefficients can be descibed by some rational polinomials of rank N. These polinomials are also multilinear in Cartan sub-algebra indices taking values from the set $I_0 = {1,2,.. N}$. The crucial point here is that for each degree one needs, in general, more than one polinomials. This in fact is related with an observation that the whole set of symmetric coefficients $ g^{A_1,A_2,.. A_p} $ is decomposed into sum subsets which are in one to one correspondence with these polinomials. We call these subsets clusters and introduce some indicators with which we specify different clusters. These indicators determine all the clusters whatever the numerical values of coefficients $g^{A_1,A_2,.. A_p}$ are. For any degree p, the number of clusters is independent of rank N. This hence allows us to generalize our results to any value of rank N. To specify the general framework explicit constructions of 4th and 5th order Casimir operators of $A_N$ Lie algebras are studied and all the polinomials which specify the numerical value of their coefficients are given explicitly.