Quasitriangular chiral WZW model in a nutshell

TL;DR

This paper provides a concise description of the quasitriangular chiral WZW model based on a specific Drinfeld double, detailing its symplectic structure and Poisson-Lie symmetry through two spectral parameter-dependent r-matrices.

Contribution

It introduces a detailed characterization of the quasitriangular chiral WZW model using two r-matrices, one trigonometric and one elliptic, for a specific affine Kac-Moody double.

Findings

Characterization of the symplectic structure via two r-matrices.

Identification of the trigonometric r-matrix with q-current algebra.

Description of the elliptic r-matrix related to field braiding.

Abstract

We give the bare-bone description of the quasitriangular chiral WZW model for the particular choice of the Lu-Weinstein-Soibelman Drinfeld double of the affine Kac-Moody group. The symplectic structure of the model and its Poisson-Lie symmetry are completely characterized by two $r$-matrices with spectral parameter. One of them is ordinary and trigonometric and characterizes the $q$-current algebra. The other is dynamical and elliptic (in fact Felder's one) and characterizes the braiding of $q$-primary fields.