Bethe subalgebras in Yangians and Kirillov-Reshetikhin crystals

TL;DR

This paper constructs affine crystal structures on spectra of Bethe subalgebras in Yangians for type A, conjectures their generalization, and proves their relation to Kirillov-Reshetikhin crystals using degeneration techniques and connections to the Feigin-Frenkel center.

Contribution

It introduces a natural affine crystal structure on spectra of Bethe subalgebras in Yangians and proves their correspondence to Kirillov-Reshetikhin crystals in type A.

Findings

Affine crystals constructed on spectra of Bethe subalgebras in type A

Degeneration of Bethe subalgebras relates to Feigin-Frenkel center

Confirmed crystals are Kirillov-Reshetikhin in type A

Abstract

Let $\mathfrak{g}$ be a complex simple finite dimensional Lie algebra and $G$ be the adjoint Lie group with the Lie algebra $\mathfrak{g}$. To every $C \in G$ one can associate a commutative subalgebra $B(C)$ in the Yangian $Y(\mathfrak{g})$, which is responsible for the integrals of the (generalized) $XXX$ Heisenberg magnet chain. Using the approach of arXiv:1708.05105, we construct a natural structure of affine crystals on spectra of $B(C)$ in Kirillov-Reshetikhin $Y(\mathfrak{g})$-modules in type $A$. We conjecture that such a construction exists for arbitrary $\mathfrak{g}$ and gives Kirillov-Reshetikhin crystals. Our main technical tool is the degeneration of Bethe subalgebras in the Yangian to commutative subalgebras $\mathcal{A}_\chi^{\mathrm{u}}$ in the universal enveloping of the current Lie algebra, $U(\mathfrak{g}[t])$, which depend on the parameter $\chi$ from the Lie algebra $\mathfrak{g}$ (and are of independent interest). We show that these subalgebras come from the Feigin-Frenkel center on the critical level as described by Feigin, Frenkel and Toledano Laredo in arXiv:math/0612798. This allows to prove that our affine crystals in type $A$ are indeed Kirillov-Reshetikhin by reducing to the crystal structure on the spectra of inhomogeneous Gaudin model which is already known (arXiv:1708.05105).