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

This paper deals with the striking fact that there is an essentially canoni cal path from the ith Lie algebra cohomology cocycle, i = 1,2,...,l, of a s imple compact Lie algebra g of rank l to the definition of its primitive Ca simir operators C-(i) of order m(i). Thus one obtains a complete set of Rac ah-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) va lid for any representation of g, and thereby to relate c((i)) to a suitably defined generalized Dynkin index. The form of the formula for c((i)) for s u(n) is known sufficiently explicitly to make clear some interesting and im portant features. For the purposes of illustration, detailed results are di splayed for some classes of representation of su(n), including all the fund amental ones and the adjoint representation. (C) 2001 American Institute of Physics.