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Free product of quantum groups
The following construction was defined by Wang in [Wan95]
Definition
Motivation for the compact case
Given two (compact) groups 
and 
, we can construct their direct product. Such a construction is generalized to the quantum case as the tensor product of quantum groups. Here, we have 
. In particular, the two factors 
and 
commute.
If we start with some pair of free compact quantum groups (such as the free orthogonal quantum group 
), we would like to generalize the direct product in a way, that the resulting C*-algebra will again be as free as possible. To achieve that, we replace the tensor product by the C*-algebraic free product.
Definition for compact quantum groups
Let and be compact quantum groups. We define their dual free product 
to be the quantum group with underlying C*-algebra 
and comultiplication the unique unital 
-homomorphism satisfying
Formally, we should rather write 
, 
, where 
and 
are the canonical inclusions into 
.
References
[(:ref:Wan95» author: Shuzhou Wang title: Free products of compact quantum groups journal: Communications in Mathematical Physics year: 1995 volume: 167 number: 3 pages: 671–692 url: http://dx.doi.org/10.1007/BF02101540 }
