Characterization of the unit object in localized quantum unipotent category
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866914156343459840 |
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| author | Matsuura, Koh Nakashima, Toshiki |
| author_facet | Matsuura, Koh Nakashima, Toshiki |
| contents | For the quiver Hecke algebra $R$, let $R\hbox{-gmod}$ be the category of finite-dimensional graded $R$-modules, and let $\widetilde{R\hbox{-gmod}[w]}$ be the localization of $R\hbox{-gmod}$. Kashiwara and the second author showed the set of equivalence classes of simple objects up to grading shifts $\mathrm{Irr}(\widetilde{R\hbox{-gmod}[w]})$ in $\widetilde{R\hbox{-gmod}[w]}$ has a crystal structure, and $\mathrm{Irr}(\widetilde{R\hbox{-gmod}[w]})$ is isomorphic to the so-called cellular crystal $\mathbb B_{\mathbf i}$. This isomorphism induces a function $\varepsilon_i^*$ on $\mathbb B_{\mathbf i}$. We give an explicit formula of $\varepsilon_i^*$, and using this formula, we give a characterization of the unit object of $\widetilde{R\hbox{-gmod}[w]}$ for the case of classical finite types. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_10157 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Characterization of the unit object in localized quantum unipotent category Matsuura, Koh Nakashima, Toshiki Representation Theory Combinatorics Quantum Algebra For the quiver Hecke algebra $R$, let $R\hbox{-gmod}$ be the category of finite-dimensional graded $R$-modules, and let $\widetilde{R\hbox{-gmod}[w]}$ be the localization of $R\hbox{-gmod}$. Kashiwara and the second author showed the set of equivalence classes of simple objects up to grading shifts $\mathrm{Irr}(\widetilde{R\hbox{-gmod}[w]})$ in $\widetilde{R\hbox{-gmod}[w]}$ has a crystal structure, and $\mathrm{Irr}(\widetilde{R\hbox{-gmod}[w]})$ is isomorphic to the so-called cellular crystal $\mathbb B_{\mathbf i}$. This isomorphism induces a function $\varepsilon_i^*$ on $\mathbb B_{\mathbf i}$. We give an explicit formula of $\varepsilon_i^*$, and using this formula, we give a characterization of the unit object of $\widetilde{R\hbox{-gmod}[w]}$ for the case of classical finite types. |
| title | Characterization of the unit object in localized quantum unipotent category |
| topic | Representation Theory Combinatorics Quantum Algebra |
| url | https://arxiv.org/abs/2511.10157 |