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$(\mathfrak{gl}_{n},\mathfrak{gl}_{m})$-duality and Olshanski homomorphism

B. Feigin, L. Rybnikov, F. Uvarov

Abstract

We show that the images of the Bethe subalgebras of the Yangians $Y(\mathfrak{gl}_{n})$ and $Y(\mathfrak{gl}_{m})$ under the homomorphisms to $U(\mathfrak{gl}_{n+m})$ given by the Olshanski centralizer construction coincide. We use this result to obtain the $(\mathfrak{gl}_{n},\mathfrak{gl}_{m})$-duality of the trigonometric Gaudin model and the XXX-spin chain. The duality is obtained in an explicit way relating the generating differential operator on one side and the generating difference operator on the other, thus agreeing with the result of Mukhin, Tarasov and Varchenko arXiv:math/0605172.

$(\mathfrak{gl}_{n},\mathfrak{gl}_{m})$-duality and Olshanski homomorphism

Abstract

We show that the images of the Bethe subalgebras of the Yangians and under the homomorphisms to given by the Olshanski centralizer construction coincide. We use this result to obtain the -duality of the trigonometric Gaudin model and the XXX-spin chain. The duality is obtained in an explicit way relating the generating differential operator on one side and the generating difference operator on the other, thus agreeing with the result of Mukhin, Tarasov and Varchenko arXiv:math/0605172.
Paper Structure (21 sections, 20 theorems, 139 equations)

This paper contains 21 sections, 20 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Preliminaries
  4. Basics on Yangians and their representations
  5. Bethe subalgebras of Yangians
  6. The duality on the level of the universal enveloping algebra $U(\mathfrak{gl}_{n+m})$
  7. Olshanski homomorphism
  8. The duality
  9. Shift of argument subalgebras
  10. The duality on the level of representations
  11. An isomorphism of Yangian modules
  12. The duality
  13. Degeneration to the duality between the XXX-type spin chain and the trigonometric Gaudin model
  14. The Bethe algebra of the quantum XXX-type spin chain
  15. The Bethe algebra of the quantum trigonometric Gaudin model
  16. ...and 6 more sections

Key Result

Lemma 3.1

Denote $C_{1,0}=(1,1,\dots, 1,0,0,\dots, 0)$ with $n$ ones and $C_{0,1}=(0,0,\dots, 0,1,1,\dots, 1)$ with $m$ ones. Then

Theorems & Definitions (35)

  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • proof
  • Remark 3.4
  • Proposition 3.5
  • Theorem 3.6
  • Proposition 4.1
  • ...and 25 more