Speaker
Description
The Yangian of a reductive Lie algebra contains a family of maximal commutative subalgebras—the Bethe subalgebras—parametrized by regular elements of the maximal torus. In the case of $\mathfrak{gl}(n)$, it is known that this family extends to a larger one indexed by points of the Deligne-Mumford compactification of $M(0,n+2)$. For any point $C$ in the real locus of this parameter space, and a fixed tame Yangian representation $V$, the Bethe subalgebra $B(C)$ acts on $V$ with simple spectrum. I will discuss the structure of the resulting unramified covering—with fiber over C given by the set of eigenlines for the action of $B(C)$, which can be identified with a collection of Gelfand-Tsetlin keystone patterns carrying a $\mathfrak{gl}(n)$-crystal structure, as well as the monodromy action realized by a type of cactus group. This is joint work with Anfisa Gurenkova and Leonid Rybnikov.
