## Final Report Summary - NCGQG (Noncommutative geometry and quantum groups)

The goal of the project was to make fundamental contributions to the study of the q-deformations of compact Lie groups and their homogeneous spaces in the operator algebraic setting and to

theory of quantum random walks. The following are the main outcomes of the project.

Development, jointly with Makoto Yamashita, of a Poisson boundary theory for monoidal categories. This theory associates a new tensor category to a given category and a probability measure on its set of simple objects, as well as a functor from the original category to the new one. The central result of the theory is a universality property of this functor. Consequences of the universality include a classification of dimension-preserving fiber functors on the q-deformations of compact Lie groups (for positive q different from one). As an application, a complete classification of non-Kac compact quantum groups of SU(n) type is obtained, answering (in the non-Kac case) a question of Woronowicz from 1988.

Construction, jointly with Makoto Yamashita, of a maximal completion of the fusion algebra of a C*-tensor category. The construction allows one to introduce property (T) for such categories and explains preservation of property (T) of quantum groups and subfactors under monoidal equivalence and isomorphism of standard invariants, respectively. Similar results were simultaneously obtained by Sorin Popa and Steffan Vaes, who also introduced other approximation properties of tensor categories in addition to property (T).

Formulation, jointly with Kenny De Commer, Lars Tuset and Makoto Yamashita, of a conjecture on the connection between two constructions of quantum symmetric spaces. One construction, due to Gail Letzter, is purely algebraic, the other is based on the theory of 2-cyclotomic KZ-equations. The conjecture should be considered as a type B analogue of the famous Kohno-Drinfeld theorem. It has been proved in the rank one case.

Computatiion, jointly with Sara Malacarne, of the Martin boundaries of the duals of free unitary quantum groups, providing the first truly new example of a computation of the noncommutative

Martin boundary since the initial computation by Sergey Neshveyev and Lars Tuset of the boundary of the dual of SUq(2) in 2002.

Development, jointly with Lars Tuset, of a general theory of deformations of C*-algebras by actions of locally compact quantum groups and measurable 2-cocycles on their duals. This theory either generalizes or provides an alternative approach to several earlier constructions, including Marc Rieffel's deformation for actions of R^n, Pawel Kasprzak's approach to Rieffel's deformation based on Magnus Landstad's theory, and the theory developed by Pierre Bieliavsky and Victor Gayral for actions of Kahlerian Lie groups.

Dynamical characterization, jointly with Makoto Yamashita, of categorical Morita equivalence for compact quantum groups. This theory can be considered as a refinement of the well-known part of the Hopf-Galois theory developed by K:-H. Ulbrich and Peter Schauenburg, which gives a characterization of monoidal equivalence of Hopf algebras.

theory of quantum random walks. The following are the main outcomes of the project.

Development, jointly with Makoto Yamashita, of a Poisson boundary theory for monoidal categories. This theory associates a new tensor category to a given category and a probability measure on its set of simple objects, as well as a functor from the original category to the new one. The central result of the theory is a universality property of this functor. Consequences of the universality include a classification of dimension-preserving fiber functors on the q-deformations of compact Lie groups (for positive q different from one). As an application, a complete classification of non-Kac compact quantum groups of SU(n) type is obtained, answering (in the non-Kac case) a question of Woronowicz from 1988.

Construction, jointly with Makoto Yamashita, of a maximal completion of the fusion algebra of a C*-tensor category. The construction allows one to introduce property (T) for such categories and explains preservation of property (T) of quantum groups and subfactors under monoidal equivalence and isomorphism of standard invariants, respectively. Similar results were simultaneously obtained by Sorin Popa and Steffan Vaes, who also introduced other approximation properties of tensor categories in addition to property (T).

Formulation, jointly with Kenny De Commer, Lars Tuset and Makoto Yamashita, of a conjecture on the connection between two constructions of quantum symmetric spaces. One construction, due to Gail Letzter, is purely algebraic, the other is based on the theory of 2-cyclotomic KZ-equations. The conjecture should be considered as a type B analogue of the famous Kohno-Drinfeld theorem. It has been proved in the rank one case.

Computatiion, jointly with Sara Malacarne, of the Martin boundaries of the duals of free unitary quantum groups, providing the first truly new example of a computation of the noncommutative

Martin boundary since the initial computation by Sergey Neshveyev and Lars Tuset of the boundary of the dual of SUq(2) in 2002.

Development, jointly with Lars Tuset, of a general theory of deformations of C*-algebras by actions of locally compact quantum groups and measurable 2-cocycles on their duals. This theory either generalizes or provides an alternative approach to several earlier constructions, including Marc Rieffel's deformation for actions of R^n, Pawel Kasprzak's approach to Rieffel's deformation based on Magnus Landstad's theory, and the theory developed by Pierre Bieliavsky and Victor Gayral for actions of Kahlerian Lie groups.

Dynamical characterization, jointly with Makoto Yamashita, of categorical Morita equivalence for compact quantum groups. This theory can be considered as a refinement of the well-known part of the Hopf-Galois theory developed by K:-H. Ulbrich and Peter Schauenburg, which gives a characterization of monoidal equivalence of Hopf algebras.