Models of quantum permutations

Abstract

For N ≥ 4 we present a series of * -homomorphisms ϕ n : C(S + N ) → 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 ∞ which defines a compact matrix quantum group S N G ⊆ S + N . We know G = S + N for N = 4, 5 from Banica's work, but we have to leave open the case N ≥ 6.

Quantum Permutation Matrices

“…In order to have more examples, let us mention a construction by Woronowicz, see for instance [49,Def. 3…”

“…Example 2. 8 The following example is taken from [49,Ex. 3.12], where p, p , q, q ∈ B(H ) may be any projections.…”

“…Example 2.9 And another one from [49,Ex. 3.13], again coming from taking the Woronowicz tensor product.…”

“…Possibly on some nice Hilbert spaces, like functional Hilbert spaces, Hardy spaces, Bergman spaces, Fock spaces? See [7,13,27,49] for some models of quantum permutation matrices.…”