By Thomas Timmermann

ISBN-10: 3037190434

ISBN-13: 9783037190432

This ebook offers an advent to the idea of quantum teams with emphasis on their duality and at the atmosphere of operator algebras. half I of the textual content offers the elemental conception of Hopf algebras, Van Daele's duality concept of algebraic quantum teams, and Woronowicz's compact quantum teams, staying in a in basic terms algebraic atmosphere. half II specializes in quantum teams within the surroundings of operator algebras. Woronowicz's compact quantum teams are handled within the atmosphere of $C^*$-algebras, and the basic multiplicative unitaries of Baaj and Skandalis are studied intimately. an overview of Kustermans' and Vaes' complete conception of in the neighborhood compact quantum teams completes this half. half III ends up in chosen subject matters, akin to coactions, Baaj-Skandalis-duality, and methods to quantum groupoids within the atmosphere of operator algebras. The publication is addressed to graduate scholars and non-experts from different fields. simply 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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B/ / for all a 2 A and b 2 B. b/ / for all a 2 A; b 2 B, and the reverse implication follows similarly. Thus, one of these conditions may be omitted in the definition of a dual pairing of Hopf -algebras. Let us consider several examples of dual pairings. 6. C// ! C/. C/ ! C/ is identified with the space of n n-matrices with vanishing trace. 14. 7. Let G be a compact connected Lie group with Lie algebra g. g/, which can be described as follows. G/. 12], and therefore every continuous finite-dimensional representation and every representative function of G is smooth.

Then the following conditions are equivalent: i) The comultiplication W A ! A ˝ A is an algebra homomorphism. ii) The multiplication m W A ˝ A ! A is a morphism of coalgebras. 7) / A ˝ A. Proof. A˝A˝A˝A id ˝† ˝ id ! A ˝ A/ respectively. 6. 7) commutes. A; / consisting of the algebra A and the comultiplication as a bialgebra. A bialgebra is called unital if it is unital as an algebra and the comultiplication is a unital algebra homomorphism; it is called counital if it is counital as a coalgebra and the multiplication is a counital morphism of coalgebras.

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. a; b/ 7! A ˚ B; A˚B /. Tensor product. Denote by A˝B the composition of the map A ˝B W A˝ Š B ! A˝A˝B ˝B with the isomorphism A˝A˝B ˝B ! A˝B ˝A˝B given by a1 ˝ a2 ˝ b1 ˝ b2 7! a1 ˝ b1 ˝ a2 ˝ b2 . A ˝ B; A˝B / is a coalgebra. If A and B possess counits A and B , respectively, then the map a ˝ b 7!

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

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