The twisted XXZ chain at roots of unity revisited

TL;DR

This paper explores the symmetries of the twisted XXZ spin-chain at roots of unity, revealing an infinite-dimensional non-abelian symmetry algebra related to the sl_2 loop algebra, with implications for integrable models.

Contribution

It explicitly constructs the non-abelian symmetry algebra for all spin sectors of the twisted XXZ chain at roots of unity, using only fundamental algebraic properties.

Findings

Identified the symmetry algebra as a Borel subalgebra of the sl_2 loop algebra.

Constructed the symmetry algebra explicitly for all spin sectors.

Used the intertwining property and Yang-Baxter relations in the proof.

Abstract

The symmetries of the twisted XXZ spin-chain (alias the twisted six-vertex model) at roots of unity are investigated. It is shown that when the twist parameter is chosen to depend on the total spin an infinite-dimensional non-abelian symmetry algebra can be explicitly constructed for all spin sectors. This symmetry algebra is identified to be the upper or lower Borel subalgebra of the sl_2 loop algebra. The proof uses only the intertwining property of the six-vertex monodromy matrix and the familiar relations of the six-vertex Yang-Baxter algebra.