Identify algebraic structures underpinning the vertex-side long-range spin chains

Identify whether the vertex-type elliptic long-range spin chains built from Baxter’s eight-vertex R-matrix—specifically the Matushko–Zotov (MZ′) chain and its undeformed Sechin–Zotov (SZ′) variant—admit extra algebraic structures analogous to those on the face side (such as affine Hecke or affine Temperley–Lieb algebra representations), and, if they do, explicitly construct these structures and clarify their role in the models’ symmetry and spectrum.

Background

The paper contrasts two landscapes of integrable long-range spin chains: the vertex-type models based on Baxter’s eight-vertex R-matrix (including the MZ′ and SZ′ chains) and the face-type models based on Felder’s dynamical R-matrix (including the q-deformed Inozemtsev chain).

On the face side, the trigonometric limit connects to affine Hecke algebras (via Baxterisation of the asymmetric six-vertex R-matrix), explaining enhanced spin symmetry and the appearance of Macdonald/Jack polynomials; the short-range limit exhibits affine Temperley–Lieb algebra representations.

The authors report that, despite structural similarities, they have not been able to identify comparable algebraic structures on the vertex side, leaving open whether analogous algebraic frameworks exist for the eight-vertex-based long-range chains.