16/12/2014, 11:00 — 12:00 — Sala P3.10, Pavilhão de Matemática
Iva Halacheva, University of Toronto
Shift of Argument Algebras and the Cactus Group
For any semisimple Lie algebra $g$, there is a family of maximal commutative subalgebras of $U(g)$, the shift of argument algebras, parametrized by regular semisimple elements. They have simple spectrum, and the fundamental group of their moduli space is the pure cactus group. In type A, the resulting monodromy action agrees with the action of the pure cactus group on crystals defined using Schutzenberger involutions. We conjecture that this is also true in general. Skew-howe duality relates this result to an analogous one for the Gaudin model of commutative subalgebras in the $n$-th tensor power of $U(g)$.
