Algebraic Quantum Permutation Groups

Abstract

Abstract. We discuss some algebraic aspects of quantum permutation groups, working over arbitrary fields. If K is any characteristic zero field, we show that there exists a universal cosemisimple Hopf algebra coacting on the diagonal algebra K n : this is a refinement of Wang's universality theorem for the (compact) quantum permutation group. We also prove a structural result for Hopf algebras having a non-ergodic coaction on the diagonal algebra K n , on which we determine the possible group gradings when K is algebraically closed and has characteristic zero.

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.…”

Untitled

“…We provide a proof of this, which is also a particular case of Dȃscȃlescu's classification [2008] (see also [Bichon 2008]). …”

“…The diagonal algebra k E is the vector space of maps from E to k with pointwise multiplication. Next we consider connected gradings of diagonal algebras [Dȃscȃlescu 2008;Bichon 2008]. The following result shows that any abelian group with the cardinality of a given set grades the diagonal algebra in a connected way, if the field contains enough roots of unity.…”

Group gradings on diagonal algebras

“…A result in this direction was proved in [8], where group gradings on the upper triangular matrix algebra are classified in the case where k is an arbitrary field. Another situation was considered by Bichon in [3], where coactions of Hopf algebras on algebras are considered from the point of view of Manin and Wang, and it is proved that there exists a Hopf algebra coaction on the diagonal algebra k n , which is universal in a large class of Hopf algebras. As an application, for algebraically closed k of characteristic zero, Bichon describes coactions of group algebras on k n , which are just group gradings on the diagonal algebra k n .…”

“…This has the advantage to solve also the prime characteristic case and the non-algebraically closed case, and also to give a more precise description of the possible groups for which there exist faithful gradings on the diagonal algebra. The approach of [3] uses the duality between group actions and gradings for finite abelian groups over algebraically closed fields of characteristic zero, and then since the algebra of automorphisms of k n is the permutation group S n , the description is done by using transitive abelian subgroups of certain permutation groups. Our description is direct and does not go through any permutation groups.…”