Coideals of Quantum Groups

We will give a brief overview of the general theory of coideals (also called coideal subalgebras) of locally compact quantum groups, with emphasis on the compact and discrete cases. In particular, we will be interested certain aspects like the role of group-like projections and the codual coideal of a coideal. The latter gives us a bijection between the coideals of a quantum group and the coideals of its dual quantum group. We will also discuss the characterization of certain classes of coideals, including compact quasi-subgroups, which are coideals that are in bijection with the (universal) idempotent states, and the coideals generated by quantum subgroups. The compact quasi-subgroups play a vital role in certain operator algebraic aspects of discrete quantum groups, in a way that is analogous to the role the structure of the subgroups of a classical group play for the structure of their reduced C*-algebras. Depending on time, we will discuss (relative) amenability and coamenability of coideals and the connection to certain properties of reduced C*-algebras. We wish to highlight open questions having to do with the tracial states and simplicity of reduced C*-algebras and dynamics of discrete quantum groups.