By Thomas Timmermann
This publication presents an advent to the speculation of quantum teams with emphasis on their duality and at the surroundings of operator algebras. half I of the textual content provides the fundamental thought of Hopf algebras, Van Daele's duality conception of algebraic quantum teams, and Woronowicz's compact quantum teams, staying in a in basic terms algebraic environment. half II specializes in quantum teams within the environment of operator algebras. Woronowicz's compact quantum teams are handled 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 conception of in the community compact quantum teams completes this half. half III results in chosen issues, reminiscent of coactions, Baaj-Skandalis-duality, and methods to quantum groupoids within the atmosphere of operator algebras. The booklet is addressed to graduate scholars and non-experts from different fields. purely uncomplicated 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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Extra resources for An invitation to quantum groups and duality
I) ) iii): The proof is similar to the proof of the implication i) ) ii). ii) ) iv): Let J Â A be a left ideal of finite codimension that is contained in ker f . Then the map W A ! b C J / WD ab C J is an algebra homomorphism. A=J / have finite dimension, the kernel I WD ker has finite codimension in A. I / D 0 implies I Â J Â ker f . iii) ) iv): Again, the proof is similar to the proof given above. vi) ) i): Let I Â A be an ideal of finite codimension that is contained in ker f . Denote by W A !
1/ ı ı SA / ˝ . A0 ; A0 / is a -coalgebra. By iii), it is also a Hopf algebra, and hence a Hopf -algebra. Š Finally, let us show that the natural isomorphism ÃA W A ! A00 of Hopf algebras is -linear. a / for all a 2 A and f 2 A0 . The following two examples explain the relation between the duality of finitedimensional Hopf algebras and the Pontrjagin duality of finite abelian groups. 2. 4. ıy / D ıx;y for all x; y 2 G. G/0 . G/0 . ıy 0 / D ıxy;z ; x 0 ;y 0 2G x 0 y 0 Dz whence "x "y D "xy . G/0 .
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.
An invitation to quantum groups and duality by Thomas Timmermann