Crystal Structure of Localized Quantum Unipotent Coordinate Category
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866909767614595072 |
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| author | Kashiwara, Masaki Nakashima, Toshiki |
| author_facet | Kashiwara, Masaki Nakashima, Toshiki |
| contents | A localized quantum unipotent coordinate category $\widetilde{\mathscr{C}_w}$ associated with a Weyl group element $w$ is a rigid monoidal category which is obtained by applying the localization process to a subcategory of the category of finite-dimensional graded modules of a quiver Hecke algebra. We shall show that the family of the isomorphism classes (up to grading shifts) of simple objects in $\widetilde{\mathscr{C}_w}$ possesses a crystal structure and it is isomorphic to the cellular crystal associated with $w$. As an application of this result, we shall show the connectedness of the crystal graph of an arbitrary cellular crystal. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_14319 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Crystal Structure of Localized Quantum Unipotent Coordinate Category Kashiwara, Masaki Nakashima, Toshiki Representation Theory Quantum Algebra 05E10, 18M05, 18M15, 16D90, 16T20, 17B37, 81R10 A localized quantum unipotent coordinate category $\widetilde{\mathscr{C}_w}$ associated with a Weyl group element $w$ is a rigid monoidal category which is obtained by applying the localization process to a subcategory of the category of finite-dimensional graded modules of a quiver Hecke algebra. We shall show that the family of the isomorphism classes (up to grading shifts) of simple objects in $\widetilde{\mathscr{C}_w}$ possesses a crystal structure and it is isomorphic to the cellular crystal associated with $w$. As an application of this result, we shall show the connectedness of the crystal graph of an arbitrary cellular crystal. |
| title | Crystal Structure of Localized Quantum Unipotent Coordinate Category |
| topic | Representation Theory Quantum Algebra 05E10, 18M05, 18M15, 16D90, 16T20, 17B37, 81R10 |
| url | https://arxiv.org/abs/2502.14319 |