Inclusions of von Neumann Algebras, and Quantum Groupoı̈ds
✍ Scribed by Michel Enock; Jean-Michel Vallin
- Publisher
- Elsevier Science
- Year
- 2000
- Tongue
- English
- Weight
- 334 KB
- Volume
- 172
- Category
- Article
- ISSN
- 0022-1236
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✦ Synopsis
From a depth 2 inclusion of von Neumann algebras M 0 /M 1 , with an operatorvalued 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 groupo@ ds. Other examples of ``quantum groupo@ ds'' are given. 2000 Academic Press G (2) =[(x, y) # G 2 ; s(x)=r( y)] with all axioms needed for associativity of this composition law, each element x having a two-side inverse x &1 , i.e., xx &1 =r(x), x &1 x=s(x) (see [R1] or [C2] for a more precise definition).