An Introduction to Yangian Symmetries

TL;DR

This paper introduces the concept of Yangian symmetries in two-dimensional quantum field theories, discussing their classical and quantum forms, algebraic structures, and related mathematical frameworks like Poisson-Lie groups.

Contribution

It provides a comprehensive review of Yangian symmetries, including classical and quantum cases, and presents new algebraic perspectives and presentations relevant to integrable models.

Findings

Yangian symmetries extend classical non-Abelian symmetries in quantum models

Introduction of alternative algebraic presentations of Yangians

Discussion of the role of Yangians in integrable quantum field theories

Abstract

We review some aspects of the quantum Yangians as symmetry algebras of two-dimensional quantum field theories. The plan of these notes is the following: 1 - The classical Heisenberg model: Non-Abelian symmetries; The generators of the symmetries and the semi-classical Yangians; An alternative presentation of the semi-classical Yangians; Digression on Poisson-Lie groups. 2 - The quantum Heisenberg chain: Non-Abelian symmetries and the quantum Yangians; The transfer matrix and an alternative presentation of the Yangians; Digression on the double Yangians. Talk given at the "Integrable Quantum Field Theories" conference held at Come, Italy , September 13-19, 1992.