# Gaudin algebras and piecewise linear transformations

SPEAKER: Leonid Rybnikov

TITLE: Gaudin algebras and piecewise linear transformations

ABSTRACT Gaudin subalgebras form a family of maximal commutative subalgebras in the N-th tensor power of a universal enveloping algebra of a semisimple Lie algebra $mathfrak{g}$. Aguirre, Felder and Veselov showed, that the closure of this family is naturally parametrized by the moduli space $bar{M}_{0,N+1}$ of stable rational curves with N+1 marked points. We describe explicitly the commutative subalgebras corresponding to boundary points of $bar{M}_{0,N+1}$. In particular, that gives rise to a quantization of the “Bending flows” integrable systems of Kapovich and Millson and describes this quantization in terms of (some generalization of) Gelfand-Tsetlin bases. In particular, this gives a covering over the real locus of $bar{M}_{0,N+1}$ whose fiber is naturally identified with the set of integral points of the Gelfand-Tsetlin polytope. We show that in the case $mathfrak{g}=sl_2$ the corresponding action of the fundamental group of the real locus of $bar{M}_{0,N+1}$ (called “cactus group”) on the fiber comes from some natural piecewise linear transformations of the Gelfand-Tsetlin polytope. I will also discuss some conjectures relating this construction to coboundary category structure on crystals.