Anne-Sophie Gleitz | Searcharxiv

arXiv2014

Quantum affine algebras at roots of unity and generalised cluster algebras

Let $U_\varepsilon^{\mathrm{res}}(L\mathfrak{sl}_2)$ be the restricted integral form of the quantum loop algebra $U_q(L\mathfrak{sl}_2)$ specialised at a root of unity $\varepsilon$. We prove that the Grothendieck ring of a tensor subcategory of representations of $U_\varepsilon^{\mathrm{res}}(L\mathfrak{sl}_2)$ is a generalised cluster algebra of type $C_{l-1}$, where $l$ is the order of $\varepsilon^2$. Moreover, we show that the classes of simple objects in the Grothendieck ring essentially coincide with the cluster monomials. We also state a conjecture for $U_\varepsilon^{\mathrm{res}}(L\mathfrak{sl}_3)$, and we prove it for $l=2$.

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arXiv2013

On the KNS Conjecture in type $E$

For the exceptional types $E_6$, $E_7$, and $E_8$, we prove that the specializations at roots of unity of the quantum dimensions of the Kirillov-Reshetikhin modules give real solutions of $\ell$-restricted $Q$-systems, as conjectured by Kuniba, Nakanishi, and Suzuki. We also show that these solutions are positive in type $E_6$. In type $E_7$ and $E_8$, we only prove positivity for a subset of the nodes of the Dynkin diagram.

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