#### Alladi Ramakrishnan Hall

#### Poisson boundaries of monoidal categories

#### Makoto Yamashita

##### Ochanomizu University

*Motivated by the theory of dimension for tensor categories by Longo and Roberts, and also by the theory of noncommutative Poisson boundary for quantum groups by Izumi, we define the notion of categorical Poisson boundary for rigid C*-tensor categories with irreducible unit. We recover many known constructions in the theory of subfactors and quantum groups as a part of this categorical boundary. In the so called weakly amenable case, we prove that the Poisson boundary has a universality property for the amenable dimension function, which has implications both to subfactors and to quantum groups. This talk is based on joint work with S. Neshveyev.*

Motivated by the theory of dimension for tensor categories by Longo and Roberts, and also by the theory of noncommutative Poisson boundary for quantum groups by Izumi, we define the notion of categorical Poisson boundary for rigid C*-tensor categories with irreducible unit. We recover many known constructions in the theory of subfactors and quantum groups as a part of this categorical boundary. In the so called weakly amenable case, we prove that the Poisson boundary has a universality property for the amenable dimension function, which has implications both to subfactors and to quantum groups. This talk is based on joint work with S. Neshveyev.

Done