Mutation of signed valued quivers and presentations of simple complex Lie algebras
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866908688339435520 |
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| author | Grant, Joseph Morigi, Davide |
| author_facet | Grant, Joseph Morigi, Davide |
| contents | We introduce a signed variant of (valued) quivers and a mutation rule that generalizes the classical Fomin-Zelevinsky mutation of quivers. To any signed valued quiver we associate a matrix that is a signed analogue of the Cartan counterpart appearing in the theory of cluster algebras. From this matrix, we construct a Lie algebra via a "Serre-like" presentation.
In the mutation Dynkin case, we define root systems using the signed Cartan counterpart and show compatibility with mutation of roots as defined by Parsons. Using results from Barot-Rivera and Pérez-Rivera, we show that mutation equivalent signed quivers yield isomorphic Lie algebras, giving presentations of simple complex Lie algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_14595 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Mutation of signed valued quivers and presentations of simple complex Lie algebras Grant, Joseph Morigi, Davide Representation Theory Rings and Algebras Primary: 17B20, Secondary: 17B22, 13F60 We introduce a signed variant of (valued) quivers and a mutation rule that generalizes the classical Fomin-Zelevinsky mutation of quivers. To any signed valued quiver we associate a matrix that is a signed analogue of the Cartan counterpart appearing in the theory of cluster algebras. From this matrix, we construct a Lie algebra via a "Serre-like" presentation. In the mutation Dynkin case, we define root systems using the signed Cartan counterpart and show compatibility with mutation of roots as defined by Parsons. Using results from Barot-Rivera and Pérez-Rivera, we show that mutation equivalent signed quivers yield isomorphic Lie algebras, giving presentations of simple complex Lie algebras. |
| title | Mutation of signed valued quivers and presentations of simple complex Lie algebras |
| topic | Representation Theory Rings and Algebras Primary: 17B20, Secondary: 17B22, 13F60 |
| url | https://arxiv.org/abs/2403.14595 |