Easy quantum groups

A closed subgroup $G{\subset }_u{U}_N^{+}$ is called easy when its associated Tannakian category $C_{kl}=Hom(u^{\otimes k},u^{\otimes l})$ appears from a category of partitions, $C=span(D)$ with $D=(D_{kl})\subset P$, via the standard implementation of partitions as linear maps. The examples abound, and the main known subgroups $G\subset U_N^{+}$ are either easy, or not far from being easy. We discuss here the basic theory, examples and known classification results for the easy quantum groups $G\subset U_N^{+}$, as well as various generalizations of the formalism, known as super-easiness theories, and the unification problem for them.