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Free bistochastic quantum group

The free bistochastic quantum groups are the elements of a sequence $(B_N^+)_{N\in \N}$ of compact matrix quantum groups introduced by Banica and Speicher in [BanSp09]. Each $B_N^+$ is a free counterpart of the bistochastic group $B_N$ of the corresponding dimension $N$ .

Definition

Given $N\in \N$ , the free bistochastic quantum group $B_N^+$ is the compact matrix quantum group $(C(B_N^+),u)$ where $u=(u_{i,j})_{i,j=1}^N$ organizes the generators $\{u_{i,j}\}_{i,j=1}^N$ of the (unital) universal C*-algebra

$C(B_N^+)\colon\hspace{-0.66em}= C^\ast_1\big\langle\{u_{i,j}\}_{i,j=1}^N\big\,\vert \,\forall_{i,j=1}^N: u_{i,j}=u_{i,j}^\ast, \, {\textstyle\sum_{k=1}^N u_{i,k}u_{j,k}={\textstyle\sum_{k=1}^N} u_{k,i}u_{k,j}=1, \, {\textstyle\sum_{k=1}^N} u_{i,k}={\textstyle\sum_{l=1}^N} u_{l,j}=1\big\rangle,$

where $1$ is the unit of the universal $C^\ast$ -algebra.

The definition can also be expressed by saying that the fundamental corpresentation matrix $u$ of $S_N^+$ is bistochastic (or doubly stochastic), which is to say that $u$ is orthogonal and each of its rows and columns sums to $1$ .

References

[BanSp09] Banica, Teodor and Speicher, Roland, 2009. Liberation of orthogonal Lie groups. Advances in Mathematics, 222, pp.1461–150.

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Free bistochastic quantum group

Definition

References

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