## Abstract

We compute the L
^{2}-Betti numbers of the free C
^{*}-tensor categories, which are the representation categories of the universal unitary quantum groups A
_{u}(F). We show that the L
^{2}-Betti numbers of the dual of a compact quantum group G(double-struck) are equal to the L
^{2}-Betti numbers of the representation category Rep(G(double-struck)) and thus, in particular, invariant under monoidal equivalence. As an application, we obtain several new computations of L
^{2}-Betti numbers for discrete quantum groups, including the quantum permutation groups and the free wreath product groups. Finally, we obtain upper bounds for the first L
^{2}-Betti number in terms of a generating set of a C
^{*}-tensor category.

Original language | English |
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Journal | Analysis & PDE |

Volume | 10 |

Issue number | 7 |

Pages (from-to) | 1757-1791 |

ISSN | 2157-5045 |

DOIs | |

Publication status | Published - 2017 |

## Keywords

- Compact quantum groups
- Discrete quantum groups
- L -Betti numbers
- Rigid C -tensor categories
- Subfactors

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