  In a recent article, we gave a definition of partition C*-algebras. These are
universal C*-algebras based on algebraic relations which are induced from
partitions of sets. In this follow up article, we show that often we can
associate a Hopf algebra structure to partition C*-algebras, and also a compact
matrix quantum group structure. This follows the lines of Banica and Speicher's
approach to quantum groups; however, we access them in a more algebraic way
circumventing Tannaka-Krein duality. We give criteria when these quantum groups
are quantum subgroups of Wang's free orthogonal quantum group. As a
consequence, we see that even if we start with (generalized) categories of
partitions which do not contain the pair partitions, in many cases we do not go
beyond the class of Banica-Speicher quantum groups (aka easy quantum groups).
However, we also discuss possible non-unitary Banica-Speicher quantum groups.
