Algebraic structures underlying the vertex-type landscape

Identify algebraic structures underlying the vertex-type long-range spin chains based on Baxter’s eight-vertex R-matrix (including the Matushko–Zotov MZ′ chain and its limits), analogous to the affine Hecke and affine Temperley–Lieb structures known on the face-type side for the q-deformed Haldane–Shastry and short-range limits.

Background

On the face-type side, the long-range limit connects to (degenerate or ordinary) affine Hecke algebras, explaining enhanced spin symmetry, while the short-range limit features representations of the affine Temperley–Lieb algebra. These structures underpin spectral properties and integrability.

The authors note they could not identify comparable algebraic structures on the vertex side. Establishing such structures for the vertex-type landscape could illuminate symmetry, degeneracies, and integrability mechanisms akin to those on the face side.