By Thomas Timmermann
This ebook offers an advent 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 thought of Hopf algebras, Van Daele's duality idea of algebraic quantum teams, and Woronowicz's compact quantum teams, staying in a in basic terms algebraic surroundings. half II specializes in quantum teams within the surroundings of operator algebras. Woronowicz's compact quantum teams are taken care of within the environment of $C^*$-algebras, and the basic multiplicative unitaries of Baaj and Skandalis are studied intimately. an overview of Kustermans' and Vaes' accomplished thought of in the neighborhood compact quantum teams completes this half. half III ends up in chosen subject matters, comparable to coactions, Baaj-Skandalis-duality, and techniques to quantum groupoids within the environment of operator algebras. The publication is addressed to graduate scholars and non-experts from different fields. merely 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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A / D a for all a; b 2 A and 2 C: A complex vector space with a fixed involution is also called a -vector space. A linear map W A ! a/ for all a 2 A. ab/ D b a for all a; b 2 A. a / D all a 2 A. A; / a -coalgebra. A -bialgebra that is a Hopf algebra is called a Hopf -algebra. 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.
A a linear map. A; /op is a Hopf algebra with antipode T . id ˝T / ı . 1/ / D Á. 1/ for all a 2 A. T ˝ id/ ı † ı . A; /cop is a Hopf algebra with antipode T . 14. i) ) ii), iii): Suppose that S is invertible. A; /cop . Á. a/// D Á. 1/ / D 22 Chapter 1. 1/ D Á. a// for all a 2 A. A; /cop be a Hopf algebra with antipode T . a// D a for all a 2 A. A; /. 16. A; /, we have S 2 D idA . Proof. A; /cop . By . 17. B; B / be Hopf algebras and F W A ! B a unital and counital morphism of bialgebras.
Now assume that G is an infinite discrete group. G/ of all functions on G is too large to carry the comultiplication . G/ of functions with finite support. G/. B/. G G/. 1. 1. Let A be an algebra. A left / right multiplier of A is a linear map T W A ! b/ for all a; b 2 A. b/a for all a; b 2 A. A/. Tl ı Sl ; Sr ı Tr /, as one can easily check. 2. For every commutative algebra A, a map T W A ! A is a left multiplier if and only if it is a right multiplier. 3. i) An algebra A is non-degenerate if (a) for every a 2 A, a ¤ 0, we have Aa ¤ 0 and aA ¤ 0, and (b) the linear span of AA is equal to A.
An invitation to quantum groups and duality by Thomas Timmermann