Finite-dimensional quantum groups of type Super A and non-semisimple modular categories
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2023
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866908863431704576 |
|---|---|
| author | Laugwitz, Robert Sanmarco, Guillermo |
| author_facet | Laugwitz, Robert Sanmarco, Guillermo |
| contents | We construct a series of finite-dimensional quantum groups as braided Drinfeld doubles of Nichols algebras of type Super A, for an even root of unity, and classify ribbon structures for these quantum groups. Ribbon structures exist if and only if the rank is even and all simple roots are odd. In this case, the quantum groups have a unique ribbon structure which comes from a non-semisimple spherical structure on the negative Borel Hopf subalgebra. Hence, the categories of finite-dimensional modules over these quantum groups provide examples of non-semisimple modular categories. In the rank-two case, we explicitly describe all simple modules of these quantum groups. We finish by computing link invariants, based on generalized traces, associated to a four-dimensional simple module of the rank-two quantum group. These knot invariants distinguish certain knots indistinguishable by the Jones or HOMFLYPT polynomials and are related to a specialization of the Links-Gould invariant. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_10685 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Finite-dimensional quantum groups of type Super A and non-semisimple modular categories Laugwitz, Robert Sanmarco, Guillermo Quantum Algebra Category Theory Representation Theory 18M15, 18M20, 17B37, 57K14, 57K16 We construct a series of finite-dimensional quantum groups as braided Drinfeld doubles of Nichols algebras of type Super A, for an even root of unity, and classify ribbon structures for these quantum groups. Ribbon structures exist if and only if the rank is even and all simple roots are odd. In this case, the quantum groups have a unique ribbon structure which comes from a non-semisimple spherical structure on the negative Borel Hopf subalgebra. Hence, the categories of finite-dimensional modules over these quantum groups provide examples of non-semisimple modular categories. In the rank-two case, we explicitly describe all simple modules of these quantum groups. We finish by computing link invariants, based on generalized traces, associated to a four-dimensional simple module of the rank-two quantum group. These knot invariants distinguish certain knots indistinguishable by the Jones or HOMFLYPT polynomials and are related to a specialization of the Links-Gould invariant. |
| title | Finite-dimensional quantum groups of type Super A and non-semisimple modular categories |
| topic | Quantum Algebra Category Theory Representation Theory 18M15, 18M20, 17B37, 57K14, 57K16 |
| url | https://arxiv.org/abs/2301.10685 |