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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2407.16812 |
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| _version_ | 1866929593051512832 |
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| author | Kim, Hyun Kyu Wang, Zhihao |
| author_facet | Kim, Hyun Kyu Wang, Zhihao |
| contents | The ${\rm SL}_3$-skein algebra $\mathscr{S}_{\bar{q}}(\mathfrak{S})$ of a punctured oriented surface $\mathfrak{S}$ is a quantum deformation of the coordinate algebra of the ${\rm SL}_3$-character variety of $\mathfrak{S}$. When $\bar{q}$ is a root of unity, we prove the Unicity Theorem for representations of $\mathscr{S}_{\bar{q}}(\mathfrak{S})$, in particular the existence and uniqueness of a generic irreducible representation. Furthermore, we show that the center of $\mathscr{S}_{\bar{q}}(\frak{S})$ is generated by the peripheral skeins around punctures and the central elements contained in the image of the Frobenius homomorphism for $\mathscr{S}_{\bar{q}}(\frak{S})$, a surface generalization of Frobenius homomorphisms of quantum groups related to ${\rm SL}_3$. We compute the rank of $\mathscr{S}_{\bar{q}}(\mathfrak{S})$ over its center, hence the dimension of the generic irreducible representation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_16812 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Unicity Theorem and the center of the ${\rm SL}_3$-skein algebra Kim, Hyun Kyu Wang, Zhihao Quantum Algebra The ${\rm SL}_3$-skein algebra $\mathscr{S}_{\bar{q}}(\mathfrak{S})$ of a punctured oriented surface $\mathfrak{S}$ is a quantum deformation of the coordinate algebra of the ${\rm SL}_3$-character variety of $\mathfrak{S}$. When $\bar{q}$ is a root of unity, we prove the Unicity Theorem for representations of $\mathscr{S}_{\bar{q}}(\mathfrak{S})$, in particular the existence and uniqueness of a generic irreducible representation. Furthermore, we show that the center of $\mathscr{S}_{\bar{q}}(\frak{S})$ is generated by the peripheral skeins around punctures and the central elements contained in the image of the Frobenius homomorphism for $\mathscr{S}_{\bar{q}}(\frak{S})$, a surface generalization of Frobenius homomorphisms of quantum groups related to ${\rm SL}_3$. We compute the rank of $\mathscr{S}_{\bar{q}}(\mathfrak{S})$ over its center, hence the dimension of the generic irreducible representation. |
| title | The Unicity Theorem and the center of the ${\rm SL}_3$-skein algebra |
| topic | Quantum Algebra |
| url | https://arxiv.org/abs/2407.16812 |