Integrability Program: Leonid Rybnikov, “Gaudin Algebras and Piecewise Linear Transformations”

Leonid Rybnikov, HSE Moscow

Title: Gaudin Algebras and Piecewise Linear Transformations
Program: Integrability in Modern Theoretical and Mathematical Physics
Speaker: Leonid Rybnikov, HSE Moscow
Date: Friday, October 5, 2012
Time: 3:00pm – 4:00pm
Place: Seminar Room 313, Simons Center

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 on the fiber comes from some natural piecewise linear transformations of the Gelfand-Tsetlin polytope.