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Toroidal Grothendieck rings and cluster algebras


We study deformations of cluster algebras with several quantum parameters, called toroidal cluster algebras, which naturally appear in the study of Grothendieck rings of representations of quantum affine algebras. In this context, we construct toroidal Grothendieck rings and we establish these are flat deformations of Grothendieck rings. We prove that for a family of monoidal categories \({\mathscr {C}}_1\) of simply-laced quantum affine algebras categorifying finite-type cluster algebras, the toroidal Grothendieck ring has a natural structure of a toroidal cluster algebra.

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  1. This is obtained from the quasi-commutation relation in (9), but with a sign change, so that it agrees with the formula in [26] corresponding to the case \( a = 0\).

  2. Here, we are using the same product as in [26], which differs from the one in [22] by replacing t with \( t^{-1}\). This change amounts to a sign change in the RHS of the formulas for \( {\mathcal {N}}_a(i,p; j,s)\).

  3. Note that in the example of Sect. 2.2 the parameters are actually denoted by \( t_1\) and \( t_2\) respectively, following the notation introduced in that section.

  4. Again, under a sign change. Here, with the purpose that the result is compatible with the quasi-commutation product between the variables \( Y_{i,p}^{\pm 1}\).

  5. Note that in [30] the authors consider the same quantum torus up to a minus sign. Moreover, the quantum Cartan matrices coincides up to a swap of rows (resp. columns) 1 and 2, and its inverse \({\widetilde{C}}(z)\) is expanded for negative z.


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The authors would like to thank the referee for his comments and remarks. The authors would like to thank also Martina Lanini, Bernard Leclerc and Bernhard Keller for discussions and references. The authors were supported by the European Research Council under the European Union’s Framework Programme H2020 with ERC Grant Agreement number 647353 Qaffine.

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Correspondence to Laura Fedele.

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Fedele, L., Hernandez, D. Toroidal Grothendieck rings and cluster algebras. Math. Z. 300, 377–420 (2022).

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