Linear coactions of discrete quantum groups on the circle
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866912905292677120 |
|---|---|
| author | Goswami, Debashish Samadder, Suchetana |
| author_facet | Goswami, Debashish Samadder, Suchetana |
| contents | For a (unital) $C^*$-algebra $\cla$, we construct a $C^*$-algebraic discrete quantum group (DQG) $\clq_{\rm aut}(\cla)$, coacting on $\cla$, which is a quantum generalization of ${\text Aut}(\cla)$ in the framework of discrete quantum groups, in the sense that any other coaction of a DQG on $\cla$ factors through the above coaction of $\clq_{\rm aut}(\cla)$. We prove by an explicit calculation that if any Kac-type $C^*$-algebraic discrete quantum group $\mathcal{Q}$ has a `weakly faithful' coaction on $C(S^1)$ which is `linear' in the sense that it leaves the space spanned by $\{ Z, \overline{Z} \}$ invariant, then $\mathcal{Q}$ must be classical, i.e. isomorphic with $C_0(Γ)$ for some discrete group $Γ$. This parallels the well-known result of non-existence of genuine compact quantum group symmetry obtained by the first author and his collaborators ([GB16] and the references therein). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_21638 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Linear coactions of discrete quantum groups on the circle Goswami, Debashish Samadder, Suchetana Quantum Algebra Operator Algebras For a (unital) $C^*$-algebra $\cla$, we construct a $C^*$-algebraic discrete quantum group (DQG) $\clq_{\rm aut}(\cla)$, coacting on $\cla$, which is a quantum generalization of ${\text Aut}(\cla)$ in the framework of discrete quantum groups, in the sense that any other coaction of a DQG on $\cla$ factors through the above coaction of $\clq_{\rm aut}(\cla)$. We prove by an explicit calculation that if any Kac-type $C^*$-algebraic discrete quantum group $\mathcal{Q}$ has a `weakly faithful' coaction on $C(S^1)$ which is `linear' in the sense that it leaves the space spanned by $\{ Z, \overline{Z} \}$ invariant, then $\mathcal{Q}$ must be classical, i.e. isomorphic with $C_0(Γ)$ for some discrete group $Γ$. This parallels the well-known result of non-existence of genuine compact quantum group symmetry obtained by the first author and his collaborators ([GB16] and the references therein). |
| title | Linear coactions of discrete quantum groups on the circle |
| topic | Quantum Algebra Operator Algebras |
| url | https://arxiv.org/abs/2508.21638 |