Universal Quantum Groups

Abstract

For each invertible m×m matrix Q a compact matrix quantum group Au(Q) is constructed. These quantum groups are shown to be universal in the sense that any compact matrix quantum group is a quantum subgroup of some of them. Their orthogonal version Ao(Q) is also constructed. Finally, we discuss related constructions in the literature.

Simple compact quantum groups I

“…[2,47,49,50]). The (noncommutative) C * -algebra of functions on the quantum group B u (Q) is generated by noncommutative coordinate functions u ij (i, j = 1, .…”

“…Let us also recall the construction of the quantum groups A u (Q) closely related to B u (Q) [47,49,50]. For every non-singular matrix Q, the quantum group A u (Q) is defined in terms of generators u ij (i, j = 1, .…”

“…symplectic leaves) in Poisson Lie group theory (see also the monograph [29] for more detailed treatment). Starting in his PhD thesis [48], the author of the present article took a different direction from the above by viewing quantum groups as intrinsic objects and found in a series of papers (including [47] in collaboration with Van Daele) several classes of compact quantum groups that cannot be obtained as deformations of Lie groups. The most important of these are the universal compact quantum groups of Kac type A u (n) and their self-conjugate counterpart A o (n) [49], the more general universal compact quantum groups A u (Q) and their self-conjugate counterpart B u (Q) [47,50], where Q ∈ GL(n, C), and the quantum automorphism groups A aut (B, tr) of finite-dimensional C * -algebras B endowed with a tracial functional tr, including the quantum permutation groups A aut (X n ) on the space X n of n points [53].…”

“…Starting in his PhD thesis [48], the author of the present article took a different direction from the above by viewing quantum groups as intrinsic objects and found in a series of papers (including [47] in collaboration with Van Daele) several classes of compact quantum groups that cannot be obtained as deformations of Lie groups. The most important of these are the universal compact quantum groups of Kac type A u (n) and their self-conjugate counterpart A o (n) [49], the more general universal compact quantum groups A u (Q) and their self-conjugate counterpart B u (Q) [47,50], where Q ∈ GL(n, C), and the quantum automorphism groups A aut (B, tr) of finite-dimensional C * -algebras B endowed with a tracial functional tr, including the quantum permutation groups A aut (X n ) on the space X n of n points [53]. 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 wreat...…”

K-amenability for amalgamated free products of amenable discrete quantum groups

“…Consider two copies S 1 ; S 2 of A o ðQÞ; the Woronowicz C Ã -algebra of an orthogonal free quantum group [21,17,5]. Let us recall that Irr C 1 and Irr C 2 can be indexed by N; with the same fusion rules as the ones of SUð2Þ: we will denote by v i;k the corresponding irreducible corepresentations, with iAf1; 2g and kAN: The discrete quantum group corresponding to A o ðQÞ has a unique non-trivial subgroup, associated to the subcategory generated by Irr D ¼ fv 2k jkANg: Its Woronowicz C Ã -algebra T is the sub-C Ã -algebra of even elements for the natural Z=2Z-grading of A o ðQÞ-when Q ¼ I n ; T is also the Woronowicz C Ã -algebra A aut ðM n Þ of [6].…”