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| Format: | Preprint |
| Published: |
2026
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| Online Access: | https://arxiv.org/abs/2601.02221 |
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| _version_ | 1866917184849051648 |
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| author | Silberberg, Lior |
| author_facet | Silberberg, Lior |
| contents | In this paper, we study the relationship between the representation theory of the quantum affine algebra $\mathcal{U}_q(\widehat{\mathfrak{sl}_\infty})$ of infinite rank, and that of the quantum toroidal algebra $\mathcal{U}_q(\mathfrak{sl}_{2n,\mathrm{tor}})$. Using monoidal categorifications due to Hernandez-Leclerc and Nakajima, we establish a cluster-theoretic interpretation of the folding map $ϕ_{2n}$ of $q$-characters, introduced by Hernandez. To this end, we introduce a notion of foldability for cluster algebras arising from infinite quivers and study a specific case of cluster algebras of type $A_\infty$. Using this interpretation of $ϕ_{2n}$, we prove a conjecture of Hernandez in new cases. Finally, we study a particular simple $\mathcal{U}_q(\mathfrak{sl}_{2n,\mathrm{tor}})$-module whose $q$-character is not a cluster variable, and conjecture that it is imaginary. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_02221 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Folding of cluster algebras and quantum toroidal algebras Silberberg, Lior Representation Theory Quantum Algebra In this paper, we study the relationship between the representation theory of the quantum affine algebra $\mathcal{U}_q(\widehat{\mathfrak{sl}_\infty})$ of infinite rank, and that of the quantum toroidal algebra $\mathcal{U}_q(\mathfrak{sl}_{2n,\mathrm{tor}})$. Using monoidal categorifications due to Hernandez-Leclerc and Nakajima, we establish a cluster-theoretic interpretation of the folding map $ϕ_{2n}$ of $q$-characters, introduced by Hernandez. To this end, we introduce a notion of foldability for cluster algebras arising from infinite quivers and study a specific case of cluster algebras of type $A_\infty$. Using this interpretation of $ϕ_{2n}$, we prove a conjecture of Hernandez in new cases. Finally, we study a particular simple $\mathcal{U}_q(\mathfrak{sl}_{2n,\mathrm{tor}})$-module whose $q$-character is not a cluster variable, and conjecture that it is imaginary. |
| title | Folding of cluster algebras and quantum toroidal algebras |
| topic | Representation Theory Quantum Algebra |
| url | https://arxiv.org/abs/2601.02221 |