By Thomas Timmermann
This booklet presents an creation to the idea of quantum teams with emphasis on their duality and at the environment of operator algebras. half I of the textual content offers the elemental idea of Hopf algebras, Van Daele's duality idea of algebraic quantum teams, and Woronowicz's compact quantum teams, staying in a basically algebraic atmosphere. half II specializes in quantum teams within the atmosphere of operator algebras. Woronowicz's compact quantum teams are handled within the surroundings of $C^*$-algebras, and the basic multiplicative unitaries of Baaj and Skandalis are studied intimately. an summary of Kustermans' and Vaes' complete conception of in the neighborhood compact quantum teams completes this half. half III results in chosen issues, corresponding to coactions, Baaj-Skandalis-duality, and techniques to quantum groupoids within the surroundings of operator algebras. The e-book is addressed to graduate scholars and non-experts from different fields. basically simple 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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Additional resources for An Invitation to Quantum Groups and Duality (Ems Textbooks in Mathematics)
A; / be a coalgebra. A linear map W A ! A; / if . 2) on page 5 commutes. A coalgebra is called counital if it has a counit. 3. B; B / is a linear map F W A ! F ˝ F / ı A . The map F is counital if A and B have counits A and B , respectively, and if B ı F D A . 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.
A morphism of -algebras/ -coalgebras/ -bialgebras/Hopf -algebras is a -linear morphism of the underlying algebras/coalgebras/bialgebras/Hopf algebras. A morphism of -algebras is also called a -homomorphism. An important class of Hopf -algebras – the class of algebraic compact quantum groups – is studied in detail in Chapter 3, and analogues of Hopf -algebras in the setting of C -algebras and von Neumann algebras are discussed in Part II. 25. i) Note the following asymmetry in the definition of -algebras and -coalgebras: for a -algebra A, the involution reverses the multiplication and can be considered as a homomorphism A !
12), respectively, are bijective. 24 Chapter 1. Hopf algebras Proof. ˝ id/ W A ˝ A ! A ˝ A; X a ˝ b 7! id ˝/ W A ˝ A ! A ˝ A; X a ˝ b 7! ˝ id/ W a ˝ b 7! T1 ˝ id/ W a ˝ b ˝ c 7! id ˝/ W a ˝ b 7! id ˝T2 / W a ˝ b ˝ c 7! 22 (). A; / be a unital bialgebra. A; / is a Hopf algebra. Proof. A; /. Let us start with the counit. a ˝ 1A // for all a 2 A. So, consider the map E W A ! A; a 7! 3. Axiomatics of Hopf algebras 25 We show that the image of E is contained in k 1A . b/ span A ˝ A. Therefore, the calculation above shows that the image of E is contained in k 1A .