Models of quantum permutations

Abstract

For $N\ge 4$ we present a series of *-homomorphisms ${\phi }_n:C(S_N^{+})\to B_n$ where $S_N^{+}$ is the quantum permutation group. They are not necessarily representations of the quantum group $S_N^{+}$ but they yield good operator algebraic models of quantum permutation matrices. The C*-algebras $B_n$ allow the construction of an inverse limit $B_{\infty }$ which defines a compact matrix quantum group $S_N⊊G\subseteq S_N^{+}$. We know $G=S_N^{+}$ for $N=4,5$ from Banica's work, but we have to leave open the case $N\ge 6$.

22 pages