Uniqueness of recovering the parameters of sectional operators on simple complex Lie algebras

Abstract

By a sectional operator on a simple complex Lie algebra g we mean a self-adjoint operator ϕ: g → g satisfying the identity [ϕx, a] = [x, b] for some chosen elements a, b ∈ g, a ≠ 0. The problem concerning the uniqueness of recovering the parameters of a given specific operator arises in many areas of geometry. The main result of the paper is as follows: if a and b are not proportional and a is regular and semisimple, then every pair of parameters p, q of the sectional operator is obtained from the pair a, b by multiplying the pair by a nonzero scalar, i.e., the parameters are recovered uniquely in a sense. It follows that the Mishchenko-Fomenko subalgebras for regular semisimple elements of the Poisson-Lie algebra coincide for proportional values of the parameters only.