An explicit construction of Casimir operators and eigenvalues. II

Abstract

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.

Group Theory Factors for Feynman Diagrams

“…None of these steps is new in itself, but we believe that in all cases we are going considerably beyond previous results (see e.g [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]). Since the application we have in mind is to Feynman diagrams, it is essential not just to develop an algorithm, but also to make sure it can be carried out efficiently.…”

“…Most of these papers, [7][8][9][10][11][12][13][14][15][16][17][18][19], give more or less explicit expressions for the Casimir eigenvalues of the classical Lie Algebras A n , B n , C n and D n and in one case, [17], also for G 2 . In [22,23] formulas for G 2 and F 4 are obtained, whereas E 8 was considered, up to order 14, in [25]. The issue of completeness of a set of Casimir operators was studied in [20,21].…”

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

“…There are however other sources of information on the subject. There is [17] where valuable explicit formulas are given for all Casimir operators of all classical algebras and also for g 2 , while [18] addresses the problem for other exceptional algebras.…”