By Thomas Timmermann
This e-book 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 offers the elemental idea 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 taken care of within the atmosphere of $C^*$-algebras, and the basic multiplicative unitaries of Baaj and Skandalis are studied intimately. an summary of Kustermans' and Vaes' entire thought of in the neighborhood compact quantum teams completes this half. half III results in chosen themes, reminiscent of coactions, Baaj-Skandalis-duality, and methods to quantum groupoids within the atmosphere of operator algebras. The e-book is addressed to graduate scholars and non-experts from different fields. merely 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 (Ems Textbooks in Mathematics)
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 ! A a linear map. A; A/, that is, if S idA D Á ı D idA S. A; A/. 18 Chapter 1. Hopf algebras Let us turn to another important convolution product. A; / is a coalgebra. 10. A; / is a bialgebra, then the maps A0 A !
G/. The map X 7! g/ ,! G//, ! 7! D! ; explicitly, the differential operator D! associated to an element ! D! 4]. Denote by e 2 G the unit element. Then the bilinear map . G/ ! D! e/; is a perfect dual pairing of Hopf algebras. Let us prove this assertion. G/, respectively. g/ ,! G/. Moreover, 34 Chapter 1. D! / for all ! g/. G/. G/. 2/ jh/ for all ! G/. 2/ . 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 !
3) for A0 commute. Let us give an example. 3), we obtain A A ÁA ı id ˝SA A Ao mA AO 0 o / A˝A 7! 3) for A0 . iv) Let us show that the involution defined in iv) turns A0 into a -algebra. We denote by W C ! C the complex conjugation. f / D ı. f ˝ g/ ı A ı ı SA D .. ı g ı ı SA / ˝ . ı f ı ı SA // ı A D g f for all f; g 2 A0 . Now, assume that A has finite dimension, and let f 2 A0 . f / D f ı mA D ı f ı ı SA ı mA D ı f ı mA ı .. ı SA / ˝ . ı SA //: 30 Chapter 1. 2/ : . 1/ ı ı SA / ˝ . A0 ; A0 / is a -coalgebra.