Doi-hopf modules over weak hopf algebras

Böhm, Gabriella

doi:10.1080/00927870008827113

Cited by 85 publications

(97 citation statements)

References 11 publications

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“…In order to prove that it is a left coideal, we need to show that it remains invariant under the right dual action of B, i.e., that (x Ia) belongs to T $ for all a # B, x # T $. The latter means that [x (1) (t Ix (2) )]=[x t] for all t # T. Applying a # B to the above identity on the left (i.e., using the right dual action of B on B* _ B=B* < B), we get…”

Section: Preliminariesmentioning

confidence: 95%

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A Galois Correspondence for II1 Factors and Quantum Groupoids

Nikshych¹,

Vaînerman²

2000

Journal of Functional Analysis

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We establish a Galois correspondence for finite quantum groupoid actions on II 1 factors and show that every finite index and finite depth subfactor is an intermediate subalgebra of a quantum groupoid crossed product. Moreover, any such subfactor is completely and canonically determined by a quantum groupoid and its coideal V-subalgebra. This allows us to express the bimodule category of a subfactor in terms of the representation category of a corresponding quantum groupoid and the principal graph as the Bratteli diagram of an inclusion of certain finite-dimensional C*-algebras related to it. Academic Press

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Section: Preliminariesmentioning

confidence: 95%

“…The unit and counit satisfy the identities =(bc (1) ) =(c (2) ) d==(bcd), (2(1) 1)(1 2(1))=(2 id) 2(1), (3) S is an anti-algebra and anti-coalgebra map such that…”

Section: Preliminariesmentioning

confidence: 99%

A Galois Correspondence for II1 Factors and Quantum Groupoids

Nikshych¹,

Vaînerman²

2000

Journal of Functional Analysis

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“…[5], [3]). Associated to a weak entwining structure (A,C, ψ R ) is the category M(ψ R ) C A of right weak entwined modules, i.e.…”

Section: C-rings Associated To Invertible Weak Entwining Structuresmentioning

confidence: 99%

The Galois Theory of Matrix C-rings

Brzeziński

Turner

2006

Appl Categor Struct

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ABSTRACT. A theory of monoids in the category of bicomodules of a coalgebra C or Crings is developed. This can be viewed as a dual version of the coring theory. The notion of a matrix ring context consisting of two bicomodules and two maps is introduced and the corresponding example of a C-ring (termed a matrix C-ring) is constructed. It is shown that a matrix ring context can be associated to any bicomodule which is a one-sided quasifinite injector. Based on this, the notion of a Galois module is introduced and the structure theorem, generalising Schneider's Theorem II [H.-J. Schneider, Israel J. Math., 72 (1990), 167-195], is proven. This is then applied to the C-ring associated to a weak entwining structure and a structure theorem for a weak A-Galois coextension is derived. The theory of matrix ring contexts for a firm coalgebra (or infinite matrix ring contexts) is outlined. A Galois connection associated to a matrix C-ring is constructed.

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“…A weak Hopf algebra is a vector space that has both algebra and coalgebra structures related to each other in a certain self-dual fashion and possesses an analogue of the linearized inverse map [3]- [5]. The main difference between ordinary and weak Hopf algebras comes from the fact that the comultiplication of the latter is no longer required to preserve the unit (equivalently, the counit is not requires to be a homomorphism) and results in the existence of two canonical subalgebras playing the role of "noncommutative bases".…”

Section: Introductionmentioning

confidence: 99%