Integrable Models Associated to Classical Representations of U_q(\widehat{sl(n)})

TL;DR

This paper constructs integrable models linked to quantum affine algebras U_q(sl(n)), generalizing the XXZ spin chain to higher ranks, and solves the n=3 case using Bethe ansatz.

Contribution

It introduces a new representation for U_q(sl(n)) based on classical representations, and derives the associated integrable Hamiltonian for any n, including explicit solutions for n=3.

Findings

Derived the R-matrix for U_q(sl(n))

Generalized the XXZ model to sl(n)

Solved the n=3 case with Bethe ansatz

Abstract

We describe a representation for $U_q(\widehat{sl(n)})$, when $q$ is not a root of unity, based on the fundamental representation of $sl(n)$. As $U_q(sl(n))$ has a Hopf algebra structure with a non-commutative co-product, we look for a intertwine matrix $R$ that relates two possible definitions of that co-product. We solve cases for $n=2$ and $n=3$, and then we generalize for any $n$. We obtain the hamiltonian associated to such matrix $R$, corresponding to a multi-state chain. As the case for $n=2$ corresponds to the XXZ model with spin $1/2$, for $n>2$ we have the generalization of the XXZ model to $sl(n)$. We show the case for $n=3$ and its solution by Bethe ansatz.