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 A1A2..Ap . It is seen that these coefficients can be described by some rational polinomials of rank N. These polinomials are also multilinear in Cartan sub-algebra indices taking values from the set I • ≡ {1, 2, .., N }. The crucial point here is that for each degree one needs, in general, more than one polinomial. This in fact is related with an observation that the whole set of symmetric coefficients g A1A2..Ap is decomposed into some sub-sets which are in one-to-one correspondence with these polinomials. 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 A1A2..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 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.
It is given a way of computing Casimir eigenvalues for Weyl orbits as well as for irreducible representations of Lie algebras. A κ(s) number of polinomials of rank N are obtained explicitly for A N Casimir operators of order s where κ(s) is the number of partitions of s into positive integers except 1. It is also emphasized that these eigenvalue polinomials prove useful in obtaining formulas to calculate weight multiplicities and in explicit calculations of the whole cohomology ring of Classical and also Exceptional Lie algebras.
We have general frameworks to obtain Poincare polynomials for Finite and also Affine types of Kac-Moody Lie algebras. Very little is known however beyond Affine ones, though we have a constructive theorem which can be applied both for finite and infinite cases. One can conclusively said that theorem gives the Poincare polynomial P(G) of a Kac-Moody Lie algebra G in the product form P(G)=P(g) R where g is a precisely chosen sub-algebra of G and R is a rational function. Not in the way which theorem says but, at least for 48 hyperbolic Lie algebras considered in this work, we have shown that there is another way of choosing a sub-algebra in such a way that R appears to be the inverse of a finite polynomial. It is clear that a rational function or its inverse can not be expressed in the form of a finite polynomial.Our method is based on numerical calculations and results are given for each and every one of 48 Hyperbolic Lie algebras.In an illustrative example however, we will give how above-mentioned theorem gives us rational functions in which case we find a finite polynomial for which theorem fails to obtain.
For a finite Lie algebra G N of rank N, the Weyl orbits W (Λ ++ ) of strictly dominant weights Λ ++
It is known that characters of irreducible representations of finite Lie algebras can be obtained by Weyl character formula including Weyl group summations which make actual calculations almost impossible except for a few Lie algebras of lower rank. By starting from Weyl character formula, we show that these characters can be re-expressed without referring to Weyl group summations. Some useful technical points are given in detail in the instructive example of G 2 Lie algebra.
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