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Freely modified bistochastic quantum group

The freely modified 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], although originally under the names $(B_N^{\prime +})_{N\in \N}$. Each $B_N^{\#+}$ can be seen as a free counterpart of the bistochastic group $B_N$ of the corresponding dimension $N$. However, differently from the other matrix groups, $B_N$ actually has two free counterparts, the second being the free modified bistochastic quantum group $B_N^{\prime+}$, defined by Weber in [Web12].

Definition

Given $N\in \N$, the freely modified 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

$$C(B_N^{\# +})\colon\hspace{-0.66em}= C^\ast_1\big\langle\{u_{i,j}\}_{i,j=1}^N\big\,\vert \,u=\overline u,\, uu^t=u^tu=I_N\otimes 1, \, \forall_{i,j=1}^n:{\textstyle\sum_{k=1}^N} u_{i,k}={\textstyle\sum_{l=1}^N} u_{l,j}\big\rangle,$$

where $\overline u=(u^\ast_{i,j})_{i,j=1}^N$ is the complex conjugate of $u$ and $u^t=(u_{j,i})_{i,j=1}^N$ the transpose, where $I_N$ is the identity $N\!\times \!N$-matrix and 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 $B_N^{\# +}$ is bistochastic#.

Note that $B_N^{\#+}$ is designated by $B_N^{\prime +}$ in [BanSp09]. It was later renamed by Weber in [Web12] after the discovery of the discovery of the easy quantum group which is now called free modified bistochastic quantum group and commonly given the symbol $B_N^{\prime +}$ instead.

Basic Properties

The fundamental corepresentation matrix $u$ of $B_N^{\# +}$ is in particular orthogonal. Hence, $B_N^{\# +}$ is a compact quantum subgroup of the free orthogonal quantum group $O_N^+$.

If $I$ denotes the closed two-sided ideal of $C(B_N^{\#+})$ generated by the relations $u_{i,j}u_{k,l}=u_{k,l}u_{i,j}$ for any $i,j,k,l=1,\ldots, N$, then $C(B_N^{\# +})/I$ is isomorphic to the $C^\ast$-algebra $C(B_N')$ of continuous functions on the modified bistochastic group $B_N'$, the subgroup $\{\pm M\,\vert\, M\in B_N\}$ of $\mathrm{GL}(N,\C)$ given by signed bistochastic matrices. Hence, $B_N^{\# +}$ is a compact quantum supergroup of $B_N'$.

Moreover, if $J$ is the closed two-sided ideal of $C(B_N^{\#+})$ generated by the relations $u_{i,j}r=ru_{i,j}$ for any $i,j=1,\ldots,N$, where $r\hspace{-0.66em}\colon= \sum_{k=1}^Nu_{1,k}$, then $C(B_N^{\# +})/J$ is isomorphic to $C(B_N^{\prime +})$, the algebra of the free modified bistochastic quantum group $B_N^{\prime +}$. Thus, in particular, $B_N^{\#+}$ is a compact quantum supergroup of $B_N^{\prime+}$.

The freely modified bistochastic quantum groups $(B_N^{\# +})_{N\in \N}$ are an easy family of compact matrix quantum groups, i.e., the intertwiner spaces of their corepresentation categories are induced by a category of partitions. More precisely, it is the category of all non-crossing partitions of even size with small blocks and even distances between legs that induces the corepresentation categories of $(B_N^{\#+})_{N\in \N}$. The partition $\singleton\otimes\singleton$ is its canonical generator.

Representation theory

Cohomology

References

[BanSp09] Banica, Teodor and Speicher, Roland, 2009. Liberation of orthogonal Lie groups. Advances in Mathematics, 222, pp.1461–150.
[Web12] Weber, Moritz, 2013. On the classification of easy quantum groups. Advances in Mathematics, 245, pp.500–533.