Groupoı̈des Quantiques Finis

Abstract

A notion of pseudo-multiplicative unitaries appeared, first with a commutative basis in the study of measured groupoids and then in a more general case in depth-2 inclusions of von Neumann algebras. This article studies the finite dimensional version of this formalism generalizing S. Baaj and G. Skandalis' works to Ž . finite dimensional quantum groupoids. Other works by T. Yamanouchi and more recently by G.

Actions and coactions of finite quantum groupoids on von Neumann algebras, extensions of the matched pair procedure

“…This is just a generalization of the demonstration of Proposition 1.3(ii) in [E1], replacing the unitary W G by the adjoint of the regular mpi defined in [Val1,4.1]. 2…”

“…It is shown in [Val1], that I is regular, if and only if it generates two C * -quantum groupoids in duality:…”

Outer actions of measured quantum groupoids

“…(iii) the finite-dimensional case had been studied by D. Nikshych and L. Vainermann [30][31][32], J.-M. Vallin [46,47] and M.-C. David [10]; in that case, non-trivial examples are given, for instance Temperley-Lieb algebras [32,10], which had appeared in subfactor theory [24].…”

“…In a finite-dimensional setting, this construction can be mostly simplified, and is studied in [30,7,8,37,[46][47][48], and examples are described. In [31], the link between these "finite quantum groupoids" and depth 2 inclusions of II 1 factors is given, and in [10] had been proved that any finite-dimensional connected C * -quantum groupoid can act outerly on the hyperfinite II 1 factor.…”

Inclusions of von Neumann algebras and quantum groupoïds III

“…In a finite-dimensional setting, this construction can be mostly simplified, and is studied in [BSz1,BSz2,NV1,Sz,Val3,Val4], and examples are described. In [NV2], the link between these "finite quantum groupoïds" and depth 2 inclusions of I I 1 factors is given.…”