CSOS models descending from chiral Potts models: Degeneracy of the eigenspace and loop algebra

TL;DR

This paper investigates the degeneracy of eigenspaces in CSOS models derived from the chiral Potts model, establishing the presence of a quantum loop algebra and explicitly determining eigenvalues.

Contribution

It demonstrates eigenspace degeneracy in these models and proves the existence of an $L({\mathfrak{sl}}_2)$ quantum loop algebra with explicit eigenvalue formulas.

Findings

Degeneracy occurs in superintegrable cases but not generally.

Existence of a quantum loop algebra in these models is established.

Eigenvalues of the models are explicitly computed.

Abstract

Monodromy matrices of the $\tau_2$ model are known to satisfy a Yang--Baxter equation with a six-vertex $R$-matrix as the intertwiner. The commutation relations of the elements of the monodromy matrices are completely determined by this $R$-matrix. We show the reason why in the superintegrable case the eigenspace is degenerate, but not in the general case. We then show that the eigenspaces of special CSOS models descending from the chiral Potts model are also degenerate. The existence of an $L({\mathfrak{sl}}_2)$ quantum loop algebra (or subalgebra) in these models is established by showing that the Serre relations hold for the generators. The highest weight polynomial (or the Drinfeld polynomial) of the representation is obtained by using the method of Baxter for the superintegrable case. As a byproduct, the eigenvalues of all such CSOS models are given explicitly.