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--- free_bistochastic_quantum_group [2020/01/02 16:44]
amang [Definition]
+++ free_bistochastic_quantum_group [2021/11/23 11:56] (current)
@@ Line 1: / Line 1: @@
 ====== Free bistochastic quantum group ======
-The **free bistochastic quantum groups** are the elements of a sequence $(B_N^+)_{N\in \N}$ of [[compact matrix quantum group|compact matrix quantum groups]] introduced by Banica and Speicher in [(:ref:BanSp09)]. Each $B_N^+$ is a [[free_orthogonal_easy_quantum_group|free]] counterpart of the [[bistochastic group]] $B_N$.
+The **free bistochastic quantum groups** are the elements of a sequence $(B_N^+)_{N\in \N}$ of [[compact matrix quantum group|compact matrix quantum groups]] introduced by Banica and Speicher in [(:ref:BanSp09)]. Each $B_N^+$ is a [[free_orthogonal_easy_quantum_group|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) [[wp>Universal_C*-algebra|universal C*-algebra]]
-$$C(B_N^+)\colon\hspace{-0.66em}= C^\ast_1\big\langle\{u_{i,j}\}_{i,j=1}^N\big\,\vert \,\forall_{i,j=1}^N: u_{i,j}=u_{i,j}^\ast, \, {\textstyle\sum_{k=1}^N u_{i,k}u_{j,k}={\textstyle\sum_{k=1}^N} u_{k,i}u_{k,j}=1, \, {\textstyle\sum_{k=1}^N} u_{i,k}={\textstyle\sum_{l=1}^N} u_{l,j}=1\big\rangle,$$
+$$C(B_N^+)\colon\hspace{-0.66em}= C^\ast_1\big\langle\{u_{i,j}\}_{i,j=1}^N\big\,\vert \,u=\overline u,\, uu^t=u^tu=I_N\otimes 1, \, \forall_{i,j=1}^N: {\textstyle\sum_{k=1}^N} u_{i,k}={\textstyle\sum_{l=1}^N} u_{l,j}=1\big\rangle,$$
-where $1$ is the unit of the universal $C^\ast$-algebra.
+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** (or **doubly stochastic**), which is to say that $u$ is orthogonal and each of its rows and columns sums to $1$.
+===== 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 [[wp>bistochastic group]] $B_N$, the subgroup of $\mathrm{GL}(N,\C)$ given by all [[wp>doubly stochastic matrix|bistochastic matrices]]. Hence, $B_N^+$ is a compact quantum supergroup of $B_N$.
+The free bistochastic quantum groups $(B_N^+)_{N\in \N}$ are an [[easy_quantum_group|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 with small blocks]] that induces the corepresentation categories of $(S_N^+)_{N\in \N}$. Its canonical generating partition is $\singleton$.
+===== Representation theory =====
+===== Cohomology =====
+===== Related quantum groups =====
-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 =====

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