Problems in the theory of quantum groups

Abstract

This is a collection of open problems in the theory of quantum groups. Emphasis is given to problems in the analytic aspects of the subject. We give a collection of problems on quantum groups that are, as far as we know, still open. Many of these problems seem to be well known, though some are new. This collection is by no means meant to contain all of the important problems in the subject. However we do believe that solutions of some of these problems will give significant contribution to the theory of quantum groups. By the nature of a collection of this kind, it would be desirable to include an extensive bibliography. Because of insufficient space, we have limited ourselves to listing only those references that are cited in the problems contained in this collection. The reader can find further references in them. 1. Classifications of compact quantum groups and representations. Analysis on quantum homogeneous spaces. Recently, three natural families of compact matrix quantum groups, Au(Q), Ao(Q) and Bu(Q), were constructed [67, 68, 66, 72, 6, 7]. These quantum groups have remarkable properties. For instance, the Au(Q)’s form a universal family of compact matrix quantum groups (namely, they are the universal analogues of the ordinary unitary groups U(n)); one can take “intersections” of these quantum groups to obtain “smaller” quantum groups (in the obvious sense); the famous quantum groups SUq(2) are special examples of the Bu(Q) by choosing an appropriate matrix Q. Unlike the quantum groups of Drinfeld-Jimbo [19, 24] and Woronowicz [81, 83], 1991 Mathematics Subject Classification: Primary 46L89, 16W30, 17B37; Secondary 22E99, 43A99, 81R50. The paper is in final form and no version of it will be published elsewhere.