The study of the torsion phenomenon for discrete quantum groups has allowed me to manipulate conveniently the corresponding homological algebra for two main purposes.
On the one hand, I have introduced a new insight for a proper formulation of the BC property for arbitrary discrete quantum groups. Moreover, this research has yield two supplementary results not contemplated in the original project. First, I have solved the "cleftness" problem for compact quantum groups, which was an open problem in the subject. Generally speaking, it states that a twisted irreducible finite-dimensional representation of a compact quantum group is equivalent to a torsion action of the quantum group. This result allows thus to study certain torsion phenomenon for compact quantum groups from a representation theoretic point of view. Second, I have realised a connection between the quantum BC property and the theory of quantum groupoids, which is innovative in the subject.
On the other hand, I have started explicit K-theory computations of C*-algebras defining relevant examples of quantum groups; namely, "quantum semi-direct products". In particular, I have given a classification of its torsion phenomenon. I have also studied the case of profinite (quantum) groups in relation with the BC property. Namely, I have showed that duals of compact groups (not necessarily connected) satisfy the BC property. This complements a previous result by R. Meyer and R. Nest. For this I have developed a reformulation of the equivariant Kasparov in terms of explicit formulae using the representation theory of a given compact quantum group. This reformulation turns out to be a key technical tool to address the general construction of compact bicrossed products in relation with approximation properties, which is now part of my future research project.
On a different note, I have defined a quantum equivariant Künneth formula using the notion of Künneth functor in order to relate the quantum BC property to the Künneth formula.
Due to their complexity, some problems have not been solved as expected. For instance, this is is the case for the problems addressed around CEP. More time and further investigation is needed. I have studied the recently discovered connections between compact quantum groups and non-local games in the framework of quantum information theory in order to address relevant questions around CEP and quantum strategies for the graph isomorphism game. In the course of the action, however, I have acquired new directions to address these problems by discovering supplementary open questions related to my original objective.
