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The free compact quantum group \(U(n)\). (Le groupe quantique compact libre \(U(n)\).) (French) Zbl 0906.17009
Consider the set \(M_n(\mathbb{C})\) of \(n\times n\) matrices over \(\mathbb{C}\). For every invertible matrix \(F\) in \(M_n(\mathbb{C})\) one can consider the universal \(C^*\)-algebra with identity, generated by elements \(\{u_{ij}\mid i,j= 1,\dots,n\}\) subject to the conditions that the matrices \(u\) and \(F\overline{u}F^{-1}\) are unitary (where \((\overline{u})_{ij}= u_{ij}^*\) for all \(i,j\)). This \(C^*\)-algebra is denoted by \(A_u(F)\). The pair \((A_u(F), u)\) is a compact matrix quantum group in the sense of S. L. Woronowicz [Commun. Math. Phys. 111, 613-665 (1987; Zbl 0627.58034)]. Moreover any compact matrix quantum group is a quantum subgroup of some \((A_u(F), u)\). These universal compact matrix quantum groups were introduced by A. Van Daele and S. Wang [Int. J. Math. 7, 255-263 (1996; Zbl 0870.17011)], where the definition is slightly more general, but the case considered here is certainly the most important one.
One of the main results in the paper (Theorem 1) concerns the representation theory of these universal quantum groups. Among other results, it is shown that the irreducible representations of the quantum group \((A_u(F),u)\) can be labeled by the free product \(\mathbb{N}* \mathbb{N}\) of two copies of \(\mathbb{N}\) in such a way that the decomposition of the tensor product of two irreducible representations into irreducible components is given by the following formula: Write \(\mathbb{N}\) in a multiplicative way and let \(\alpha\) and \(\beta\) denote the generators of the two copies of \(\mathbb{N}\) in \(\mathbb{N}* \mathbb{N}\). There is a unique antimutiplicative involution \(x\to \overline{x}\) such that \(\overline{\alpha}= \beta\). Then, if \(r_x\) denotes the representation labeled by \(x\), one has \[ r_x\otimes r_y= \sum_{a,b,c} r_{ab}, \] where the sum is taken over all \(a,b,c\) in \(\mathbb{N}* \mathbb{N}\) such that \(x= ac\) and \(y= \overline{c}b\). Together with the fact that \(r\alpha= u\) and \(r_\beta= \overline{u}\), this determines the dual of \((A_u(F),u)\).
In Theorem 2 of the paper, the converse of this result is also shown: If the irreducible representations of a compact matrix quantum group \((A,u)\) are labeled by \(\mathbb{N}* \mathbb{N}\) in such a way that \(r_e= 1\) (where \(e\) denotes the identity in \(\mathbb{N}* \mathbb{N}\)) and \(r_\alpha= u\) and \(r_\beta= \overline{u}\) and the tensor product of two irreducible representations decomposes as above, then \((A,u)\) is of the form \((A_u(F),u)\) for some \(F\).
There are many more interesting results in the paper. Some of them relate with the theory of free random variables as introduced by D. Voiculescu [see Prog. Math. 92, 45-60 (1990; Zbl 0744.46055)]. The above fusion rule is e.g. related with the properties of Voiculescu’s circular variable. Another result is the simplicity of the reduced \(C^*\)-algebras \(A_u (F)_{\text{red}}\), obtained by taking the regular representation of \((A_u(F),u)\).

17B37 Quantum groups (quantized enveloping algebras) and related deformations
46L89 Other “noncommutative” mathematics based on \(C^*\)-algebra theory
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
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