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

## TL;DR

This paper explores how Bethe subalgebras in the Yangian of rak{gl}_n degenerate, revealing a deep connection with the Deligne-Mumford moduli space and providing explicit descriptions of these degenerations.

## Contribution

It establishes that the degenerations of Bethe subalgebras are parametrized by the Deligne-Mumford space and describes their structure and composition, extending to other root systems.

## Key findings

- Closure of Bethe subalgebras corresponds to rak{M}_{0,n+2}
- All degenerations are compositions of simplest degenerations
- Explicit descriptions of the simplest degenerations

## 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.