Weak Hopf Algebras and Reducible Jones Inclusions of Depth 2. I: From Crossed products to Jones towers

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dc.creatorNill, Florian
dc.creatorSzlachanyi, Kornel
dc.creatorWiesbrock, Hans-Werner
dc.date1998-06-23
dc.date.accessioned2026-07-07T05:25:10Z
dc.date.available2026-07-07T05:25:10Z
dc.descriptionWe apply the theory of finite dimensional weak C^*-Hopf algebras A as developed by G. Böhm, F. Nill and K. Szlachányi to study reducible inclusion triples of von-Neumann algebras N \subset M \subset (M\cros\A). Here M is an A-module algebra, N is the fixed point algebra and \M\cros\A is the crossed product extension. ``Weak'' means that the coproduct Δon A is non-unital, requiring various modifications of the standard definitions for (co-)actions and crossed products. We show that acting with normalized positive and nondegenerate left integrals l\in\A gives rise to faithful conditional expectations E_l: M-->N, where under certain regularity conditions this correspondence is one-to-one. Associated with such left integrals we construct ``Jones projections'' e_l\in\A obeying the Jones relations as an identity in M\cros\A. Finally, we prove that N\subset M always has finite index and depth 2 and that the basic Jones construction is given by the ideal M_1:=M e_l M \subset M\cros\A, where under appropriate conditions M_1 = M\cros\A. In a subsequent paper we will show that converseley any reducible finite index and depth-2 Jones tower of von-Neumann factors (with finite dimensional centers) arises in this way.
dc.descriptionLatex, 63 pages
dc.identifierhttps://arxiv.org/abs/math/9806130
dc.identifierhttp://arxiv.org/abs/math/9806130
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/77083
dc.subjectQuantum Algebra
dc.subjectMathematical Physics
dc.titleWeak Hopf Algebras and Reducible Jones Inclusions of Depth 2. I: From Crossed products to Jones towers
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