The spin-chain equivalence conjecture for the infinite Temperley–Lieb algebra

Let $w_{0},w_{1}$ be link states, let $\mathcal{S}(w_{0})$ and $\mathcal{S}(w_{1})$ be the associated spin-chain representations, and define

$$\mathcal{F}_{w_{0},w_{1}}: \operatorname{Rep} U_{q}(\mathfrak{sl}_{2})\longrightarrow \operatorname{Rep} TL_{\infty}$$

by $\mathcal{F}_{w_{0},w_{1}}(-):=\operatorname{Hom}(-,\mathcal{S}(w_{0})\oplus\mathcal{S}(w_{1}))$. Let $\mathcal{C}(w_{0},w_{1})$ be the Serre subcategory generated by finitely generated $TL_{\infty}(q+q^{-1})$ representations generated by link states that differ from $w_{0}$ or $w_{1}$ at only finitely many points. Spin-chain equivalence conjecture. The functor $\mathcal{F}_{w_{0},w_{1}}$ defines an equivalence of abelian categories between the category of finite dimensional $U_{q}(\mathfrak{sl}_{2})$ representations and $\mathcal{C}(w_{0},w_{1})$.

This conjecture predicts that the finite-dimensional representation theory of $U_{q}(\mathfrak{sl}_{2})$ is categorically realized by the specified Serre subcategory of representations of the infinite Temperley–Lieb algebra. The supplied text gives no evidence that the conjecture has been proved or refuted.