By Thomas Timmermann
This e-book offers an creation to the idea of quantum teams with emphasis on their duality and at the atmosphere of operator algebras. half I of the textual content provides the fundamental conception of Hopf algebras, Van Daele's duality conception of algebraic quantum teams, and Woronowicz's compact quantum teams, staying in a merely algebraic atmosphere. half II makes a speciality of quantum teams within the environment of operator algebras. Woronowicz's compact quantum teams are taken care of within the surroundings of $C^*$-algebras, and the basic multiplicative unitaries of Baaj and Skandalis are studied intimately. an summary of Kustermans' and Vaes' accomplished thought of in the community compact quantum teams completes this half. half III results in chosen issues, similar to coactions, Baaj-Skandalis-duality, and methods to quantum groupoids within the surroundings of operator algebras. The booklet is addressed to graduate scholars and non-experts from different fields. basically uncomplicated wisdom of (multi-) linear algebra is needed for the 1st half, whereas the second one and 3rd half suppose a few familiarity with Hilbert areas, $C^*$-algebras, and von Neumann algebras.
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Extra resources for An invitation to quantum groups and duality
A; / is a coalgebra and the comultiplication is understood, we freely speak of A itself as a coalgebra. 2. i) Every coalgebra has at most one counit. idA ˝ 2/ ıD. 1 ˝ 2/ ıD 2 ı. B; B /, we can construct the following new coalgebras: Coopposite coalgebra. Denote by † W A ˝ A ! A ˝ A the flip map a ˝ b 7! b ˝ a. A; A /. Evidently, a linear map W A ! A; A /cop . A; A /, that is, if † ı A D A . Direct sum. Denote by A˚B the composition of the map A ˚B W A˚B ! B ˝ B/ ,! A ˚ B/. A ˚ B; A˚B / is a coalgebra.
F; a/ 7! f a turns A into a left module over A0 . id ˝ h/ ı W A ! A. 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.
G/0 . ıy 0 / D ıxy;z ; x 0 ;y 0 2G x 0 y 0 Dz whence "x "y D "xy . G/0 . "z / D "z ˝ "z . "z / D 1 for all z 2 G. "x / D "x 1 . 4. G/0 ! kG given by "x 7! 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.
An invitation to quantum groups and duality by Thomas Timmermann