An Invitation to Quantum Groups and Duality (Ems Textbooks - download pdf or read online

By Thomas Timmermann

ISBN-10: 3037190434

ISBN-13: 9783037190432

This booklet offers an advent to the speculation of quantum teams with emphasis on their duality and at the atmosphere of operator algebras. half I of the textual content provides the fundamental concept of Hopf algebras, Van Daele's duality thought of algebraic quantum teams, and Woronowicz's compact quantum teams, staying in a in simple terms algebraic surroundings. half II makes a speciality of quantum teams within the surroundings of operator algebras. Woronowicz's compact quantum teams are taken care of within the environment of $C^*$-algebras, and the elemental multiplicative unitaries of Baaj and Skandalis are studied intimately. an summary of Kustermans' and Vaes' accomplished idea of in the community compact quantum teams completes this half. half III ends up in chosen subject matters, equivalent to coactions, Baaj-Skandalis-duality, and ways to quantum groupoids within the surroundings of operator algebras. The ebook is addressed to graduate scholars and non-experts from different fields. basically easy wisdom of (multi-) linear algebra is needed for the 1st half, whereas the second one and 3rd half imagine a few familiarity with Hilbert areas, $C^*$-algebras, and von Neumann algebras.

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A; B/ becomes an algebra. A; k/ into an algebra. When we speak of A0 as an algebra, we always refer to this algebra structure. 8. 1. A; B/ need not be unital. A; B/. 2/ / D f . A; B/ and a 2 A. iii) Every algebra homomorphism F W B ! A; B/ ! A; C /; f 7! D; D / is a coalgebra and G W D ! A; B/ ! D; B/; g 7! g ı G; is an algebra homomorphism. A; /; B/ 7! F; G/ 7! F ı G D G ı F define a bifunctor from the categories of coalgebras and algebras to the category of algebras. 9. A; / be a unital and counital bialgebra and S W A !

29. The antipode of a Hopf -algebra is -linear if and only if it is involutive in the sense that S 2 D id. Proof. S ı / ı . 4 The duality of Hopf algebras The concept of a bialgebra and of a Hopf algebra has an intrinsic symmetry that gives rise to a duality which can be considered as an analogue of Pontrjagin duality. In the finite-dimensional case, this duality behaves very nicely. In the infinitedimensional case, additional concepts and stronger assumptions are needed, and a satisfying duality theory will only be achieved in Chapter 2.

Z/. 5. 2. kG/0 vanishes on an ideal I Â kG that has finite codimension. Uy C I / WD Uxy C I . Denote by 2 V 0 the map given by Ux C I 7! x/. Ux C I / D . G/. x/ D . x/v/, where is a representation of G on some finite-dimensional vector space V and 2 V 0 , v 2 V . kG/0 corresponding to f vanishes on the kernel of the representation Q W kG ! V /, Ux 7! V /. 13. 8 i). 7]. 14. G/ ! 4. 1]. 1/ D 0g [1, p. 198]. 6. Chapter 2 Multiplier Hopf algebras and their duality A multiplier Hopf algebra is a non-unital generalization of a Hopf algebra, where the target of the comultiplication is no longer the twofold tensor product of the underlying algebra, but an enlarged multiplier algebra.

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An Invitation to Quantum Groups and Duality (Ems Textbooks in Mathematics) by Thomas Timmermann


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