5 research outputs found
Quantum affine algebras at roots of unity and generalised cluster algebras
Let be the restricted
integral form of the quantum loop algebra specialised
at a root of unity . We prove that the Grothendieck ring of a
tensor subcategory of representations of
is a generalised cluster
algebra of type , where is the order of . 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
, and we prove it for .Comment: 26 pages, 9 figure
Generalised cluster algebras and -characters at roots of unity
International audienceShapiro and Chekhov (2011) have introduced the notion of generalised cluster algebra; we focus on an example in type . On the other hand, Chari and Pressley (1997), as well as Frenkel and Mukhin (2002), have studied the restricted integral form of a quantum affine algebra where is a root of unity. Our main result states that the Grothendieck ring of a tensor subcategory of representations of is a generalised cluster algebra of type , where is the order of . We also state a conjecture for , and sketch a proof for .Shapiro et Chekhov (2011) ont introduit la notion d'algèbre amassée généralisée; nous étudions un exemple en type . Par ailleurs, Chari et Pressley (1997), ainsi que Frenkel et Mukhin (2002), ont étudié la forme entière restreinte d'une algèbre affine quantique où est une racine de l'unité. Notre résultat principal affirme que l'anneau de Grothendieck d'une sous-catégorie tensorielle de représentations de est une algèbre amassée généralisée de type , où est l'ordre de . Nous conjecturons une propriété similaire pour et donnons un aperçu de la preuve pour
Cluster scattering diagrams and theta functions for reciprocal generalized cluster algebras
We give a construction of generalized cluster varieties and generalized
cluster scattering diagrams for reciprocal generalized cluster algebras, the
latter of which were defined by Chekhov and Shapiro. These constructions are
analogous to the structures given for ordinary cluster algebras in the work of
Gross, Hacking, Keel, and Kontsevich. As a consequence of these constructions,
we are also able to construct theta functions for generalized cluster algebras,
again in the reciprocal case, and demonstrate a number of their structural
properties.Comment: Updated to reflect reviewers' comments and to use the preferred
journal formattin
Queensland University of Technology: Handbook 1996
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