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Quantum groupoids, their representat...

Nikshych, Dmitri Alexandrovich.

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Quantum groupoids, their representation categories, symmetries ofvon Neumann factors, and dynamical quantum groups.

Record Type:	Electronic resources : Monograph/item
Title/Author:	Quantum groupoids, their representation categories, symmetries ofvon Neumann factors, and dynamical quantum groups./
Author:	Nikshych, Dmitri Alexandrovich.
Description:	151 p.
Notes:	Source: Dissertation Abstracts International, Volume: 62-02, Section: B, page: 0889.
Contained By:	Dissertation Abstracts International62-02B.
Subject:	Mathematics. -
Online resource:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=3005942
ISBN:	0493151125

Quantum groupoids, their representation categories, symmetries ofvon Neumann factors, and dynamical quantum groups.
Nikshych, Dmitri Alexandrovich.

Quantum groupoids, their representation categories, symmetries ofvon Neumann factors, and dynamical quantum groups. - 151 p.

Source: Dissertation Abstracts International, Volume: 62-02, Section: B, page: 0889.

Thesis (Ph.D.)--University of California, Los Angeles, 2001.

Finite quantum groupoids (weak Hopf algebras) were recently introduced by G. Bohm and K. Szlachanyi as a generalization of usual Hopf algebras and groupoid algebras. 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. The main difference between weak and ordinary 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 required to be a homomorphism) and results in the existence of two canonical subalgebras playing the role of "non-commutative bases". Weak Hopf algebras provide a suitable algebraic framework for studying various non-commutative phenomena that cannot be otherwise described by means of Hopf algebras, such as, e.g., finite depth subfactors with non-integer index and semisimple monoidal categories with non-integer quantum dimensions.

ISBN: 0493151125Subjects--Topical Terms:

515831
Mathematics.

Quantum groupoids, their representation categories, symmetries ofvon Neumann factors, and dynamical quantum groups.
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020 $a 0493151125
035 $a (UnM)AAI3005942
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040 $a UnM $c UnM
100 1 $a Nikshych, Dmitri Alexandrovich. $3 1903916
245 1 0 $a Quantum groupoids, their representation categories, symmetries ofvon Neumann factors, and dynamical quantum groups.
300 $a 151 p.
500 $a Source: Dissertation Abstracts International, Volume: 62-02, Section: B, page: 0889.
500 $a Chair: Edward G. Effros.
502 $a Thesis (Ph.D.)--University of California, Los Angeles, 2001.
520 $a Finite quantum groupoids (weak Hopf algebras) were recently introduced by G. Bohm and K. Szlachanyi as a generalization of usual Hopf algebras and groupoid algebras. 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. The main difference between weak and ordinary 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 required to be a homomorphism) and results in the existence of two canonical subalgebras playing the role of "non-commutative bases". Weak Hopf algebras provide a suitable algebraic framework for studying various non-commutative phenomena that cannot be otherwise described by means of Hopf algebras, such as, e.g., finite depth subfactors with non-integer index and semisimple monoidal categories with non-integer quantum dimensions.
520 $a We describe the foundations of the theory of weak Hopf algebras and illustrate it by motivating examples. We develop the theory of monoidal categories of representations of weak Hopf algebras and show that their additional symmetry properties, such as braiding, ribbon and modular structures, can be used for constructing Reshetikhin-Turaev type invariants of links and 3-manifolds.
520 $a We investigate the relation between weak Hopf algebras and dynamical quantum groups and show that the result of a dynamical twisting of an ordinary finite-dimensional Hopf algebra is a weak Hopf algebra. We use this observation to introduce and study dynamical quantum groups at roots of unity.
520 $a Finally, we study the algebraic structure of weak Hopf C*-algebras and explain how they characterize finite index depth 2 inclusions of II 1 factors. We prove a Galois correspondence for arbitrary finite depth subfactors that allows to describe them as certain "quotients" of weak Hopf C*-algebra crossed products.
590 $a School code: 0031.
650 4 $a Mathematics. $3 515831
650 4 $a Physics, General. $3 1018488
690 $a 0405
690 $a 0605
710 2 0 $a University of California, Los Angeles. $3 626622
773 0 $t Dissertation Abstracts International $g 62-02B.
790 1 0 $a Effros, Edward G., $e advisor
790 $a 0031
791 $a Ph.D.
792 $a 2001
856 4 0 $u http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=3005942