Graphs having no quantum symmetry

Abstract

We consider circulant graphs having p vertices, with p prime. To any such graph we associate a certain number k, that we call type of the graph. We prove that for p >> k the graph has no quantum symmetry, in the sense that the quantum automorphism group reduces to the classical automorphism group.

Quantum groups acting on 4 points

“…Is there an analogue of Burnside's theorem for quantum permutation groups? A related question was studied in [7].…”

“…The first one is the construction of non-classical quantum permutation groups, i.e. quantum subgroups of Q n , using finite graphs [14,15,4,5,6,7]. The other direction is the study of the structure of the C * -algebra C(Q n ).…”

Simple compact quantum groups I

“…Further studies of these quantum groups reveal remarkable properties: (1) According to deep work of Banica [2][3][4], the representation rings (also called the fusion rings) of the quantum groups B u (Q) (when QQ is a scalar) are all isomorphic to that of SU(2) (see [2,Théorème 1]), and the representation rings of A aut (B, τ ) (when dim(B) 4, τ being the canonical trace on B) are all isomorphic to that of SO(3) (see [4,Theorem 4.1]), and the representation ring of A u (Q) is almost a free product of two copies of Z (see [3,Théorème 1]); (2) The compact quantum groups A u (Q) admit ergodic actions on both finite and infinite injective von Neumann factors [54]; (3) The special A u (Q)'s for positive Q and B u (Q)'s for Q satisfying the property QQ = ±I n are classified up to isomorphism using respectively the eigenvalues of Q (see [56,Theorem 1.1]) and polar decomposition of Q and eigenvalues of |Q| (see [56,Theorem 2.4]), and the general A u (Q)'s and B u (Q)'s for arbitrary Q have explicit decompositions as free products of the former special ones (see [56, Theorems 3.1, 3.3 and Corollaries 3.2, 3.4]); (4) Certain quantum symmetry groups in the theory of subfactors were found by Banica [6,7] to fit in the theory of compact quantum groups; (5) The quantum permutation groups A aut (X n ) admit interesting quantum subgroups that appear in connection with other areas of mathematics, such as the quantum automorphism groups of finite graphs and the free wreath products discovered by Bichon [15,16]. See also [17] and [8][9][10][11][12][13][14] and the references therein for other interesting results related to the quantum permutation groups.…”

Algebraic Quantum Permutation Groups

“…A s (n, Z/pZ) could be used in some contexts (especially in the context of quantum automorphism groups of finite graphs as in [5]), but this idea has not been fruitful yet.…”