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 environment of operator algebras. half I of the textual content provides the fundamental thought of Hopf algebras, Van Daele's duality concept of algebraic quantum teams, and Woronowicz's compact quantum teams, staying in a only algebraic atmosphere. half II specializes in quantum teams within the atmosphere of operator algebras. Woronowicz's compact quantum teams are handled within the environment of $C^*$-algebras, and the basic multiplicative unitaries of Baaj and Skandalis are studied intimately. an overview of Kustermans' and Vaes' entire concept of in the neighborhood compact quantum teams completes this half. half III results in chosen subject matters, similar to coactions, Baaj-Skandalis-duality, and techniques to quantum groupoids within the surroundings of operator algebras. The publication is addressed to graduate scholars and non-experts from different fields. purely simple 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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G/ for all f; g 2 A0 . a; f / 7! a f turns A into a right module over A0 . 11. A; / is a Hopf algebra, then S a D Á. a// D a S for all a 2 A. 3 Properties of the antipode The antipode of a Hopf algebra satisfies several fundamental relations that are not obvious from the definition. To some extent, the antipode of a Hopf algebra behaves like the inversion of a group: the inversion of a group is antimultiplicative, and the antipode of a Hopf algebra is both antimultiplicative and anticomultiplicative.

15) with f D gh, or the Leibniz rule for higher order derivatives of the product gh, it is easy to see that the equation above remains valid if we replace X 2 g by an element ! g/. exp. tX // D . 4. /jf / for all ! g/. v) Let ! D! G/. We show that then ! D 0. Since the operator D! G/ is closed under left translation, it follows that D! G// D 0. We show that then also D! G// D 0, and this implies ! D 0. G/ carries a natural locally convex topology, and D! is continuous with respect to this topology [32, Chapter XVII].

Ux for all x 2 G is an isomorphism of Hopf algebras. 3. Let G be a finite abelian group. F Ux /. / WD CG. Moreover, the Fourier transform F W CG ! 8, a special case of the isomorphism CG ! G/. A ˝ A/0 is strict. mA /0 W A0 ! A ˝ A/0 . This problem can be addressed in several ways. A ˝ A/ that is larger than A ˝ A. In the following sections, we discuss the first and second approach; the third one involves several additional concepts and is presented in Chapter 2. A; A / be a finite-dimensional bialgebra.

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

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