A Galois Correspondence for II1 Factors and Quantum Groupoids

Abstract

We establish a Galois correspondence for finite quantum groupoid actions on II 1 factors and show that every finite index and finite depth subfactor is an intermediate subalgebra of a quantum groupoid crossed product. Moreover, any such subfactor is completely and canonically determined by a quantum groupoid and its coideal V-subalgebra. This allows us to express the bimodule category of a subfactor in terms of the representation category of a corresponding quantum groupoid and the principal graph as the Bratteli diagram of an inclusion of certain finite-dimensional C*-algebras related to it. Academic Press

On the Structure of Weak Hopf Algebras

“…We refer the reader to the original article [1] and recent survey [15] for a detailed introduction to the theory of weak Hopf algebras and its applications.…”

“…for all h ∈ H. The images of the counital maps (14) are separable subalgebras of H, called target and source bases or counital subalgebras of H. These subalgebras commute with each other; moreover…”

“…This appears to be a natural property, since it is satisfied by dynamical quantum groups [4] and weak Hopf algebras arising as symmetries of Jones-von Neumann subfactors [13], [14].…”

“…For the foundations of the weak Hopf algebra theory we refer the reader to [1] and to the recent survey [15].…”

“…[21]. An example of an indecomposable semisimple weak Hopf algebra of dimension 13 having such properties was given in [2] (see also [14,Appendix]). …”