Inclusions of von Neumann Algebras, and Quantum Groupoı̈ds

From a depth 2 inclusion of von Neumann algebras M0 ⊂M1 , with an operator-valued weight verifying a regularity condition, we construct a pseudo-multiplicative unitary, which leads to two structures of Hopf bimodules, dual to each other. Moreover, we construct an action of one of these structures on the algebra M1 such that M0 is the fixed point subalgebra, the algebra M2 given by the basic construction being then isomorphic to the crossed-product. We construct on M2 an action of the other structure, which can be considered as the dual action. If the inclusion M0 ⊂M1 is irreducible, we recover quantum groups, as proved in former papers. This construction generalizes the situation which occurs for actions (or co-actions) of groupoı̈ds. Other examples of “quantum groupoı̈ds” are given.