  This is an introduction to the quantum groups, or rather to the simplest
quantum groups. The idea is that the unitary group $U_N$ has a free analogue
$U_N^+$, whose standard coordinates $u_{ij}\in C(U_N^+)$ are allowed to be
free, and the closed subgroups $G\subset U_N^+$ can be thought of as being the
compact quantum Lie groups. There are many interesting examples of such quantum
groups, for the most designed in order to help with questions in quantum
mechanics and statistical mechanics, and some general theory available as well,
including Peter-Weyl theory, Tannakian duality, Brauer theorems and Weingarten
integration. We discuss here the basic aspects of all this.
