Baxter Q-Operators and Representations of Yangians

TL;DR

This paper introduces a novel approach connecting Baxter Q-operators to Yangians, providing an algebraic framework for solving integrable spin chains and deriving Bethe equations without traditional ansatz methods.

Contribution

It establishes a new algebraic method linking Baxter Q-operators with Yangians and harmonic oscillator solutions, enabling systematic solutions of spin chains.

Findings

Derived the hierarchy of functional equations for transfer matrices and Q-operators.

Provided an algebraic derivation of nested Bethe equations.

Introduced degenerate solutions of the Yang-Baxter equation related to harmonic oscillators.

Abstract

We develop a new approach to Baxter Q-operators by relating them to the theory of Yangians, which are the simplest examples for quantum groups. Here we open up a new chapter in this theory and study certain degenerate solutions of the Yang-Baxter equation connected with harmonic oscillator algebras. These infinite-state solutions of the Yang-Baxter equation serve as elementary, "partonic" building blocks for other solutions via the standard fusion procedure. As a first example of the method we consider sl(n) compact spin chains and derive the full hierarchy of operatorial functional equations for all related commuting transfer matrices and Q-operators. This leads to a systematic and transparent solution of these chains, where the nested Bethe equations are derived in an entirely algebraic fashion, without any reference to the traditional Bethe ansatz techniques.