Shift of argument subalgebras in Poisson algebrasand their quantization

Abstract:

The symmetric algebra S(g) of a Lie algebra g carries a natural Poisson bracket. Shift of argument subalgebras (introduced by Fomenko and Mishchenko in 1978) form a family of maximal Poisson-commutative subalgebras in S(g) for semisimple g. This family is parametrized by regular elements of the dual space g*. I will discuss the quantization problem for shift of argument subalgebras, namely, how to lift these subalgebras to commutative subalgebras in the universal enveloping algebra U(g), and how to describe the spectra of the "quantum shift of argument subalgebras" of U(g) on (finite-dimensional) g-modules. These questions are related to the classical representation theory, in particular, it was observed by Vinberg, that the Gelfand-Tsetlin subalgebra in U(gl_n) is a certain limit of quantum shift of argument subalgebras, and hence the spectra of quantum shift of argument subalgebras on a finite-dimensional g-module can be regarded as a deformation of the corresponding Gelfand-Tsetlin polytope.