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

TL;DR

This paper establishes a duality between certain algebraic structures in integrable models using the Olshanski homomorphism, connecting Bethe subalgebras of Yangians with explicit operator relations.

Contribution

It proves the coincidence of Bethe subalgebra images under Olshanski homomorphisms and derives explicit duality relations for the Gaudin model and XXX-spin chain.

Findings

Bethe subalgebras images coincide under Olshanski homomorphisms

Explicit duality between differential and difference operators

Connection with Mukhin, Tarasov, and Varchenko's results

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.