Limits of Gaudin algebras, quantization of bending flows, Jucys--Murphy elements and Gelfand--Tsetlin bases

TL;DR

This paper explores the limits of Gaudin algebras, introduces new commutative subalgebras related to bending flows and Gelfand--Tsetlin bases, and proves spectrum simplicity in certain Gaudin models.

Contribution

It introduces new commutative subalgebras as limits of Gaudin algebras and connects them to bending flows and Gelfand--Tsetlin bases, advancing understanding of Gaudin models.

Findings

New commutative subalgebras obtained as limits of Gaudin algebras

Established connection between these subalgebras, bending flows, and Gelfand--Tsetlin bases

Proved spectrum simplicity in specific Gaudin model cases

Abstract

Gaudin algebras form a family of maximal commutative subalgebras in the tensor product of $n$ copies of the universal enveloping algebra $U(\g)$ of a semisimple Lie algebra $\g$. This family is parameterized by collections of pairwise distinct complex numbers $z_1,...,z_n$. We obtain some new commutative subalgebras in $U(\g)^{\otimes n}$ as limit cases of Gaudin subalgebras. These commutative subalgebras turn to be related to the hamiltonians of bending flows and to the Gelfand--Tsetlin bases. We use this to prove the simplicity of spectrum in the Gaudin model for some new cases.