Algebraic structure of classical integrability for complex sine-Gordon

TL;DR

This paper fully characterizes the algebraic structure of the classical r-matrix in the complex sine-Gordon model, revealing a deformed Yang-Baxter structure with implications for quantization.

Contribution

It introduces a detailed algebraic framework involving matrices a and s, and explores deformations and quantization issues of the complex sine-Gordon model.

Findings

Algebraic structure characterized by matrices a and s.

Deformation of classical reflection and Yang-Baxter equations.

Discussion on quantization of the algebraic structure.

Abstract

The algebraic structure underlying the classical $r$-matrix formulation of the complex sine-Gordon model is fully elucidated. It is characterized by two matrices $a$ and $s$, components of the $r$ matrix as $r=a-s$. They obey a modified classical reflection/Yang--Baxter set of equations, further deformed by non-abelian dynamical shift terms along the dual Lie algebra $su(2)^*$. The sign shift pattern of this deformation has the signature of the twisted boundary dynamical algebra. Issues related to the quantization of this algebraic structure and the formulation of quantum complex sine-Gordon on those lines are introduced and discussed.