Bethe subalgebras in Yangians and the wonderful compactification

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 $\mathfrak{g}$.

arXiv comment21 pages, a mistake in the statement of Proposition 2.29 is corrected, the proof of Proposition 2.29 is extended

References in corpus (10)

• Quantum Groups and Quantum Cohomology
• Yangians and quantizations of slices in the affine Grassmannian
• Bethe Subalgebras in Twisted Yangians
• On a class of representations of the Yangian and moduli space of monopoles
• Feigin-Frenkel center in types B, C and D
• Equivalences between three presentations of orthogonal and symplectic Yangians
• On the wonderful compactification
• The $R$-matrix presentation for the Yangian of a simple Lie algebra
• Highest weights for truncated shifted Yangians and product monomial crystals
• Braid group actions, Baxter polynomials, and affine quantum groups

Cited by in corpus (3)

• Braid group actions, Baxter polynomials, and affine quantum groups
• Spectra of Bethe subalgebras of $Y({\mathfrak{gl}}_n)$ in tame representations
• Simplicity of spectra for Bethe subalgebras in $Y({\mathfrak{gl}}_2)$