Bialgebroid actions on depth two extensions and duality

A general notion of depth two for ring homomorphism N→ M is introduced. The step two centralizers A=End N M N and B=(M⊗ NM) N in the Jones tower above N→ M are shown in a natural way via H-equivalence to be dual bimodules for Morita equivalent endomorphism rings, the step one and three centralizers,...

Full description

Saved in:
Bibliographic Details
Published in:Advances in mathematics (New York. 1965) 2003-10, Vol.179 (1), p.75-121
Main Authors: Kadison, Lars, Szlachányi, Kornél
Format: Article
Language:English
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:A general notion of depth two for ring homomorphism N→ M is introduced. The step two centralizers A=End N M N and B=(M⊗ NM) N in the Jones tower above N→ M are shown in a natural way via H-equivalence to be dual bimodules for Morita equivalent endomorphism rings, the step one and three centralizers, R= C M ( N) and C=End N– M ( M⊗ N M). We show A and B to possess dual left and right R-bialgebroid structures which generalize Lu's fundamental bialgebroids over an algebra. There are actions of A and B on M and E′= End NM with Galois properties. If M | N is depth two and Frobenius with R a separable algebra, we show that A and B are dual weak Hopf algebras fitting into a duality-for-actions tower extending previous results in this area for subfactors and Frobenius extensions.
ISSN:0001-8708
1090-2082
DOI:10.1016/S0001-8708(02)00028-2