Quantum cluster algebras and representations of shifted quantum affine algebras
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914622352654336 |
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| author | Paganelli, Francesca |
| author_facet | Paganelli, Francesca |
| contents | We construct a new quantization $K_t(\mathcal{O}^{sh}_{\mathbb{Z}})$ of the Grothendieck ring of the category $\mathcal{O}^{sh}_{\mathbb{Z}}$ of representations of shifted quantum affine algebras (of simply-laced type). We establish that our quantization is compatible with the quantum Grothendieck ring $K_t(\mathcal{O}^{\mathfrak{b},+}_{\mathbb{Z}})$ for the quantum Borel affine algebra, namely that there is a natural embedding $K_t(\mathcal{O}^{\mathfrak{b},+}_{\mathbb{Z}})\hookrightarrow K_t(\mathcal{O}^{sh}_{\mathbb{Z}})$. Our construction is partially based on the cluster algebra structure on the classical Grothendieck ring discovered by Geiss-Hernandez-Leclerc. As first applications, we formulate a quantum analogue of $QQ$-systems (that we make completely explicit in type $A_1$). We also prove that the quantum oscillator algebra is isomorphic to a localization of a subalgebra of our quantum Grothendieck ring and that it is also isomorphic to the Berenstein-Zelevinsky's quantum double Bruhat cell $\mathbb{C}_t[SL_2^{w_0,w_0}]$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_05008 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantum cluster algebras and representations of shifted quantum affine algebras Paganelli, Francesca Representation Theory Quantum Algebra We construct a new quantization $K_t(\mathcal{O}^{sh}_{\mathbb{Z}})$ of the Grothendieck ring of the category $\mathcal{O}^{sh}_{\mathbb{Z}}$ of representations of shifted quantum affine algebras (of simply-laced type). We establish that our quantization is compatible with the quantum Grothendieck ring $K_t(\mathcal{O}^{\mathfrak{b},+}_{\mathbb{Z}})$ for the quantum Borel affine algebra, namely that there is a natural embedding $K_t(\mathcal{O}^{\mathfrak{b},+}_{\mathbb{Z}})\hookrightarrow K_t(\mathcal{O}^{sh}_{\mathbb{Z}})$. Our construction is partially based on the cluster algebra structure on the classical Grothendieck ring discovered by Geiss-Hernandez-Leclerc. As first applications, we formulate a quantum analogue of $QQ$-systems (that we make completely explicit in type $A_1$). We also prove that the quantum oscillator algebra is isomorphic to a localization of a subalgebra of our quantum Grothendieck ring and that it is also isomorphic to the Berenstein-Zelevinsky's quantum double Bruhat cell $\mathbb{C}_t[SL_2^{w_0,w_0}]$. |
| title | Quantum cluster algebras and representations of shifted quantum affine algebras |
| topic | Representation Theory Quantum Algebra |
| url | https://arxiv.org/abs/2507.05008 |