Inclusions of Von Neumann Algebras and Quantum Groupoı̈ds II

By Michel Enock. 2000

65 citations recorded by the source corpus.

Abstract From a depth 2 inclusion of von Neumann algebras M 0 ⊂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 M 1 such that M 0 is the fixed point subalgebra, the algebra M 2 given by the basic construction being then isomorphic to the crossed-product. We construct on M 2 an action of the other structure, which can be considered as the dual action. If the inclusion M 0 ⊂ M 1 is irreducible, we recover quantum groups, as proved in former papers. This construction generalizes the situation which occurs for actions (or co-actions) of groupoids. Other examples of “quantum groupoids” are given.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f26943d23167808892e0ac82b71145f57fc4fe57

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Finite Quantum Groupoids and Their Applications (2000, methodology): “versa. A suitable functional analytic framework for studying quantum groupoids (not necessarily ﬁnite) in the spirit of the S. Baaj-G. Skandalis multiplicative unitaries [BS] was developed in [EVal], [E2] in connection with depth 2 inclusions of von Neumann algebras. A closely related notion of a Ho”
Galois objects and cocycle twisting for locally compact quantum groups (2008, background): “For further properties of the fiber product, see [7].”