Casimir eigenvalues for universal Lie algebra

Abstract

For two different natural definitions of Casimir operators for simple Lie algebras we show that their eigenvalues in the adjoint representation can be expressed polynomially in the universal Vogel's parameters $\alpha, \beta, \gamma$ and give explicit formulae for the generating functions of these eigenvalues.Comment: Slightly revised versio

X 2 series of universal quantum dimensions

“…Westbury [8] found a universal formula for quantum dimension of the adjoint representation. Sergeev, Veselov and one of the present authors derived [9] a universal formula for generating function for the eigenvalues of higher Casimir operators on the adjoint representation.…”

Universal formula for the Hilbert series of minimal nilpotent orbits

“…where the product is taken over positive roots such that (θ, α ∨ ) = 0. It would be interesting to deduce from here our formula (13), which is a universal form of the right hand side of (16). Let us consider as an example the A n -case, when X = P (O min ) is the hyperplane section of the Segre variety [7,8].…”

“…Let us consider as an example the A n -case, when X = P (O min ) is the hyperplane section of the Segre variety [7,8]. The formula (13) gives in this case…”

On universal knot polynomials

“…The Rosso-Jones expression and its universalization closely resembles Okubo's formula [95] for eigenvalues of higher order Casimirs, used in [96] to obtain universal expression for generating function of eigenvalues of higher Casimir operators.…”