Representations of at roots of unity and generalised cluster algebras

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• 作者：Anne-Sophie Gleitz ^{gleitz@math.unistra.fr" class="auth_mail" title="E-mail the corresponding author}
• 刊名：European Journal of Combinatorics
• 出版年：2016
• 出版时间：October 2016
• 年：2016
• 卷：57
• 期：Complete
• 页码：94-108
• 全文大小：783 K

文摘

We show that the Grothendieck rings of finite-dimensional representations of the quantum loop algebra of sl₂${\mathfrak{s}\mathfrak{l}}_2$ at roots of unity have the combinatorial structure of a generalised cluster algebra of type C$C$. 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 $U_{\epsilon }^{\mathrm{res}}\left(L{\mathfrak{s}\mathfrak{l}}_3\right)$, and prove it when the root of unity is of order 2.