2018年度RIMS共同研究(グループ型)

作用素環論とテンソル圏論の最近の進展

1月21日--23日@RIMS 111

  • Schedule

    1月21日(月) 1月22日(火) 1月23日(水)
    9:30--10:30
    紅村冬大(F. Komura) 守山貴顕(T. Moriyama)
    10:45--11:45
    山下真(M. Yamashita) A. Marrakchi
    13:30--14:30 山下真(M. Yamashita) 山下真(M. Yamashita)
    14:45--15:45 荒野悠輝(Y. Arano) 荒野悠輝(Y. Arano) Program in pdf
    16:00--17:00 A. Marrakchi A. Marrakchi

  • Titles and Abstracts
    Y. Arano Introduction to complex quantum groups

    I will overview how the quantum group techniques can be applied to the subfactors. First we observe that the classification of subfactors can be interpreted as a classification of actions of tensor categories, which can be seen as a slight generalization of actions of quantum groups and I will present some results on this direction. Then I explain how the representation theory of complex quantum groups appears to show approximation properties of tensor categories, which is important in such classification.
    F. Komura Quotients of étale groupoids and the abelianizations of groupoid C*-algebras

    The study of C*-algebras associated to étale groupoids, known to be groupoid C*-algebras, was initiated by Renault in 1980. It is a natural task to characterize the property of groupoid C*-algebras by the language of étale groupoids. In this talk, we introduce quotients of étale groupoids, which often become non-Hausdorff. Then, we apply quotients of étale groupoids to analysis of the ideal structure and the abelianizations of groupoid C*-algebras.
    A. Marrakchi Full factors, bicentralizer flow and approximately inner automorphisms

    We show that a factor M is full if and only if the C*-algebra generated by its left and right regular representations contains the compact operators. We prove that the bicentralizer flow of a type III$_1$ factor is always ergodic. As a consequence, for any type III$_1$ factor M and any $\lambda\in]0,1]$, there exists an irreducible AFD type III$_\lambda$ subfactor with expectation in M. Moreover, any type III$_1$ factor M which satisfies $M\cong M\otimes R_\lambda$ for some $\lambda\in]0,1]$ has trivial bicentralizer. Finally, we give a counter-example to the characterization of approximately inner automorphisms conjectured by Connes and we prove a weaker version of this conjecture. In particular, we obtain a new proof of Kawahigashi-Sutherland-Takesaki's result that every automorphism of the AFD type III$_1$ factor is approximately inner.
    T. Moriyama Weak density of orbit equivalence classes and free products of infinite abelian groups

    It has been known that for a countable group, the structure of the space of probability -measure-preserving (p.m.p.) actions reflects many properties of the group. In this talk, we will show that if a group is the free product of infinite abelian groups, then for every free p.m.p. action of the group, its orbit equivalence class is weakly dense in the space of p.m.p. actions. This extends Lewis Bowen's result for free groups.
    M. Yamashita Quantum symmetry through module categories

    This lecture is an introduction to the duality between symmetry of algebraic structures and tensor categories, which is a very powerful guiding principle in mathematics and mathematical physics. A protoptype of this principle is the famous Tannaka-Krein duality for compact groups. In the more recent development, this manifested in many different ways, including the standard invariant of subfactors, structure of quantum field theory, and classification of quantum homogeneous spaces. I will try to explain where module categories arise, and how to "see" them through algebraic structures such as algebra objects in tensor categories, and Poisson bracket on homogeneous spaces. Plan of lectures: + Basics on categorical duality - (C*-)tensor categories - Ostrik’s correspondence between algebra objects in a tensor category and module categories + Braided module categories - representation category of $q$-deformation quantum groups - module categories from cyclotomic Knizhnik-Zamolodchikov equations + Dynamical r/R-matrices - Poisson homogeneous spaces naturally appear as quasi-classical limit of module categories