No abstract

From an irreducible depth 2 inclusion of factors, verifying a regularity condition, we construct a multiplicative unitary, and an action, at every level of the canonical tower constructed from the inclusion; when this inclusion admits a faithful semifinite normal operator-valued weight, stronger conditions are given, and the tower appears then as a crossed-product construction. In particular we rederive Herman and Ocneanu's results when the inclusion admits a faithful normal conditional expectation, and the tower is then the crossed-product construction, alternatively by a compact quantum group and by its dual, and, more precisely, according to Yamagami's result, by a compact type Kac algebra and by its dual. AcademicPress, Inc.

In a former article, in collaboration with Jean-Michel Vallin, we have constructed two "quantum groupoïds" dual to each other, from a depth 2 inclusion of von Neumann algebras M 0 ⊂ M 1 , in such a way that the canonical Jones'tower associated to the inclusion can be described as a tower of successive crossed-products by these two structures. We are now investigating in greater details these structures in the presence of an appropriate modular theory on the basis M 0 ∩ M 1 , and we show how these examples fit with Lesieur's "measured quantum groupoïds".

Some examples of quantum groups in literature arise as deformations of a locally compact group by a "dual" 2-cocycle. We make this construction in the framework of Kac algebras; we show that these deformations are still Kac algebras; using this construction, we give new quantizations of the Heisenberg group. From this point of view, we analyse the dimension 8 non-trivial example of Kac and Paljutkin, and give a new example of non-trivial dimension 12 semi-simple *-Hopf algebras (a dimension 12 Kac algebra).