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Bethe subalgebras in Yangians and the wonderful compactification

Aleksei Ilin, Leonid Rybnikov

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 $\mathfrak{g}$.

Bethe subalgebras in Yangians and the wonderful compactification

Abstract

We study the family of Bethe subalgebras in the Yangian parameterized by the corresponding adjoint Lie group . We describe their classical limits as subalgebras in the algebra of polynomial functions on the formal Lie group . 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 and describe the subalgebras corresponding to generic points of any stratum in as Bethe subalgebras in the Yangian of the corresponding Levi subalgebra in .

Paper Structure

This paper contains 26 sections, 33 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Bethe subalgebras in the Yangian
  3. Wonderful closure.
  4. Restriction to the maximal torus.
  5. $RTT$-presentaton of the Yangian
  6. The paper is organized as follows.
  7. Acknowledgements.
  8. Three realizations of the Yangian
  9. Notation
  10. The algebra $\mathcal{O}(G_1[[t^{-1}]])$
  11. Filtration on $Y_{new}({\mathfrak{g}})$
  12. Filtration on $Y_{V}({\mathfrak{g}})$
  13. Relation between two filtrations on $Y({\mathfrak{g}})$
  14. ABC generators.
  15. Yangian of Levi subalgebra
  16. ...and 11 more sections

Key Result

Theorem 2.4

(kamn) Ordered monomials in the variables $e_{\alpha}^{(r)}, h_i^{(r)}, f_{\alpha}^{(r)}$, with $\alpha\in\Phi^+,\ i\in\Delta,\ r\in\mathbb{Z}_{>0}$, form a PBW basis of $Y_{new}({\mathfrak{g}})$.

Theorems & Definitions (74)

  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Remark 2.5
  • Proposition 2.6
  • proof
  • Theorem 2.7
  • Theorem 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 64 more