Uniqueness of Recovering the Parameters of Sectional Operators on Simple Complex Lie Algebras

A. Yu. Konyaev

Matematičeskie zametki, Tome 90 (2011) no. 3, pp. 384-393

Résumé

By a sectional operator on a simple complex Lie algebra $\mathfrak g$ we mean a self-adjoint operator $\phi\colon\mathfrak g\to\mathfrak g$ satisfying the identity $[\phi x,a]=[x,b]$ for some chosen elements $a,b\in\mathfrak g$, $a\ne0$. 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.

Détail

Keywords: simple complex Lie algebra, sectional operator, caustic, Mishchenko–Fomenko algebra, Killing form, root system, Weyl basis, Jacobi identity.
Mots-clés : semi-simple element of a Poisson–Lie algebra, Cartan subalgebra