An explicit construction of Casimir operators and eigenvalues. I

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^A1,A2,...,Ap. It is seen that these coefficients can be described by some rational polynomials of rank N. These polynomials are also multilinear in Cartan subalgebra indices taking values from the set I₀ = {1,2,...,N}. The crucial point here is that for each degree one needs, in general, more than one polynomial. This in fact is related to an observation that the whole set of symmetric coefficients g^A1,A2,...,Ap is decomposed into some sub-sets which are in one-to-one correspondence with these polynomials. We call these sub-sets clusters and introduce some indicators with which we specify different clusters. These indicators determine all the clusters whatever the numerical values of coefficients g^A1,A2,...,Ap 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 contructions of fourth and fifth order Casimir operators of A_N Lie algebras are studied and all the polynomials which specify the numerical value of their coefficients are given explicitly.