Deformed Cartan matrices and generalized preprojective algebras II: General type
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866911770475495424 |
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| author | Fujita, Ryo Murakami, Kota |
| author_facet | Fujita, Ryo Murakami, Kota |
| contents | We propose a definition of deformed symmetrizable generalized Cartan matrices with several deformation parameters, which admit a categorical interpretation by graded modules over the generalized preprojective algebras in the sense of Geiß-Leclerc-Schröer. Using the categorical interpretation, we deduce a combinatorial formula for the inverses of our deformed Cartan matrices in terms of braid group actions. Under a certain condition, which is satisfied in all the symmetric cases or in all the finite and affine cases, our definition coincides with that of the mass-deformed Cartan matrices introduced by Kimura-Pestun in their study of quiver $\mathcal{W}$-algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_14315 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Deformed Cartan matrices and generalized preprojective algebras II: General type Fujita, Ryo Murakami, Kota Representation Theory Quantum Algebra 16G20, 17B37, 16W50, 17B67, 81R50 We propose a definition of deformed symmetrizable generalized Cartan matrices with several deformation parameters, which admit a categorical interpretation by graded modules over the generalized preprojective algebras in the sense of Geiß-Leclerc-Schröer. Using the categorical interpretation, we deduce a combinatorial formula for the inverses of our deformed Cartan matrices in terms of braid group actions. Under a certain condition, which is satisfied in all the symmetric cases or in all the finite and affine cases, our definition coincides with that of the mass-deformed Cartan matrices introduced by Kimura-Pestun in their study of quiver $\mathcal{W}$-algebras. |
| title | Deformed Cartan matrices and generalized preprojective algebras II: General type |
| topic | Representation Theory Quantum Algebra 16G20, 17B37, 16W50, 17B67, 81R50 |
| url | https://arxiv.org/abs/2302.14315 |