Extension of a Borel subalgebra symmetry into the sl(2) loop algebra symmetry for the twisted XXZ spin chain at roots of unity and the Onsager algebra

TL;DR

This paper proves that finite-dimensional highest weight representations of a Borel subalgebra can be extended to the full sl(2) loop algebra in the context of the twisted XXZ spin chain at roots of unity, supporting a conjecture about its symmetry structure.

Contribution

It establishes a mathematical result extending Borel subalgebra representations to the full sl(2) loop algebra, providing evidence for the conjectured symmetry of the twisted transfer matrix.

Findings

Finite-dimensional highest weight representations extend to sl(2) loop algebra.

Supports the conjecture of sl(2) loop algebra symmetry at roots of unity.

Links to Onsager algebra symmetry in related models.

Abstract

We discuss a conjecture that the twisted transfer matrix of the six-vertex model at roots of unity with some discrete twist angles should have the sl(2) loop algebra symmetry. As an evidence of this conjecture, we show the following mathematical result on a subalgebra of the sl(2) loop algebra, which we call a Borel subalgebra: any given finite-dimensional highest weight representation of the Borel subalgebra is extended into that of the sl(2) loop algebra, if the parameters associated with it are nonzero. Thus, if operators commuting or anti-commuting with the twisted transfer matrix of the six-vertex model at roots of unity generate the Borel subalgebra, then they also generate the sl(2) loop algebra. The result should be useful for studying the connection of the sl(2) loop algebra symmetry to the Onsager algebra symmetry of the superintegrable chiral Potts model.