Quantum groups, from a functional analysis perspective

Abstract

‎It is well-known that any compact Lie group appears as closed subgroup of a unitary group‎, ‎$G\subset U_N$‎. ‎The unitary group $U_N$ has a free analogue $U_N^+$‎, ‎and the study of the closed quantum subgroups $G\subset U_N^+$ is a problem of general interest‎. ‎We review here the basic tools for dealing with such quantum groups‎, ‎with all the needed preliminaries included‎, ‎and we discuss as well a number of more advanced topics‎.