Degeneration of Bethe subalgebras in the Yangian of ${\mathfrak{gl}}_n$

Abstract

We study degenerations of Bethe subalgebras $B(C)$ in the Yangian $Y({\mathfrak{gl}}_n)$, where $C$ is a regular diagonal matrix. We show that closure of the parameter space of the family of Bethe subalgebras, which parametrizes all possible degenerations, is the Deligne-Mumford moduli space of stable rational curves $\overline{M_{0,n+2}}$. All subalgebras corresponding to the points of $\overline{M_{0,n+2}}$ are free and maximal commutative. We describe explicitly the "simplest" degenerations and show that every degeneration is the composition of the simplest ones. The Deligne-Mumford space $\overline{M_{0,n+2}}$ generalizes to other root systems as some De Concini-Procesi resolution of some toric variety. We state a conjecture generalizing our results to Bethe subalgebras in the Yangian of arbitrary simple Lie algebra in terms of this De Concini-Procesi resolution.

19 pages, Lett Math Phys (2017)

References in corpus (2)

• Bethe Subalgebras in Twisted Yangians
• Gaudin subalgebras and stable rational curves