An Introduction to Yangian Symmetries

TL;DR

This work surveys Yangian symmetries in two-dimensional integrable systems, tracing the emergence of non-Abelian loop-algebra structures from the classical Heisenberg model and lifting them to the quantum setting in the Heisenberg chain. It links nonlocal charges and their Poisson (and Hopf) algebra to the transfer/m Monodromy matrix formalism, showing that the first two charges $Q^0$ and $Q^1$ generate the full (semi-classical) and quantum Yangians, respectively. The text develops both classical (Poisson) and quantum (R-matrix) frameworks, including the quantum determinant and comultiplication, and culminates with the double Yangian as a complete deformation of the loop algebra, all within the context of quantum integrable systems. Additionally, it situates these structures within Poisson–Lie and dressing transformations, highlighting the geometric underpinnings of Yangian symmetries and their role in exact solvability via algebraic methods such as the Bethe Ansatz.

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.