Bethe subalgebras in Braided Yangians and Gaudin-type models

TL;DR

This paper introduces Bethe subalgebras within braided Yangians linked to involutive or Hecke symmetries, demonstrating their commutativity and application to Gaudin-type models.

Contribution

It establishes the commutativity of quantum symmetric polynomials in braided Yangians and constructs associated Bethe subalgebras with applications to integrable models.

Findings

Quantum symmetric polynomials commute with each other.

Bethe subalgebras are generated within braided Yangians.

Application to Gaudin-type models and integrable systems.

Abstract

In \cite{GS1} the notion of braided Yangians of Reflection Equation type was introduced. Each of these algebras is associated with an involutive or Hecke symmetry $R$. Besides, the quantum analogs of certain symmetric polynomials (elementary symmetric ones, power sums) were suggested. In the present paper we show that these quantum symmetric polynomials commute with each other and consequently generate a commutative Bethe subalgebra. As an application, we get some Gaudin-type models and the corresponding Bethe subalgebras.