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The free unitary compact quantum group

Teodor Banica
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Abstract

The free analogues of $U(n)$ in Woronowicz's compact quantum group theory are the quantum groups $\{A_u(F)|F\in GL(n,\mathbb C)\}$ introduced by Van Daele and Wang. We classify here their irreducible representations. Their fusion rules turn to be related to the combinatorics of Voiculescu's circular variable. If $F\bar{F}\in\mathbb R I_n$ we find an embedding $A_u(F)_{red}\subset C(\mathbb T)*_{red}A_o(F)$, where $A_o(F)$ is the deformation of $SU(2)$ that we previously studied. We use the representation theory and Powers' method for showing that the reduced algebras $A_u(F)_{red}$ are simple, with at most one trace.

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Teodor Banica. 2017-11-22. The free unitary compact quantum group. https://arxiv.org/abs/math/9901042

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