Bethe subalgebras in Yangians and the wonderful compactification

TL;DR

This paper explores Bethe subalgebras in Yangians, describing their classical limits, algebraic structure, and extension to the wonderful compactification, revealing their relation to Levi subalgebras.

Contribution

It introduces a new family of Bethe subalgebras parameterized by the adjoint Lie group and extends their study to the wonderful compactification, linking to Levi subalgebras.

Findings

Bethe subalgebras are free polynomial algebras for regular parameters.

Classical limits of these subalgebras are described as polynomial functions on formal Lie groups.

Extension of Bethe subalgebras to the compactification relates to Levi subalgebras.

Abstract

We study the family of Bethe subalgebras in the Yangian $Y(\mathfrak{g})$ parameterized by the corresponding adjoint Lie group $G$. We describe their classical limits as subalgebras in the algebra of polynomial functions on the formal Lie group $G_1[[t^{-1}]]$. In particular we show that, for regular values of the parameter, these subalgebras are free polynomial algebras with the same Poincare series as the Cartan subalgebra of the Yangian. Next, we extend the family of Bethe subalgebras to the De Concini--Procesi wonderful compactification $\overline{G}\supset G$ and describe the subalgebras corresponding to generic points of any stratum in $\overline{G}$ as Bethe subalgebras in the Yangian of the corresponding Levi subalgebra in $\mathfrak{g}$.