Comments:

247 pages, year 2003, 8 chs., intermediate to advanced level operator (commutative and non-commutative) algebra, and beyond

Abstract:

Two major papers by Leonid Vainerman, one with Stefaan Vaes, and the other his editorial, algebraically detailed, Introduction provide the background and advances in the field of locally compact quantum groups up to 2002. The previous work on Kac algebras is also explained and the locally compact quantum groups are defined in terms of C*-- or von Neumann algebras equipped with a co-associative multiplication and also associated left- and right-Haar measures defined by two semi-finite normal weights. The remaining five papers deal respectively with: Quantum groupoids (by Michael Enock, pp. 17 to 48; Peter Schoenburg, pp.79 to 104; Galois actions by finite QG's by Korne'l Szlachanyi, pp 106-126; J. M. Vallin, isometries and QG's, pp. 229), plus two chapters on specific cases of quantum groups. The concluding chapter by Alfons Van Deele is on Multiplier Hopf *-algebras representations of locally compact quantum groups (pp.230-247).\\\\ Editorial Book Description verbatim quote:\\ "This volume contains seven papers written by participants of the 69th meeting of theoretical physicists and mathematicians held in Strasbourg (February 21-23, 2002)... The book contains seven refereed research papers on locally compact quantum groups and groupoids by leading experts in the respective fields. These contributions are based on talks presented on the occasion of the meeting between mathematicians and theoretical physicists held in Strasbourg from February 21 to February 23, 2002. Topics covered are: various constructions of locally compact quantum groups and their multiplicative unitaries; duality theory for locally compact quantum groups; combinatorial quantization of flat connections associated with SL(2,c); quantum groupoids, especially coming from Depth 2 Extensions of von Neumann algebras, C*-algebras and Rings. Many mathematical results are motivated by problems in theoretical physics. Historical remarks set the results presented in perspective. ///// Directed at research mathematicians and theoretical physicists as well as graduate students, the volume will give an overview of a field of research in which great progress has been achieved in the last few years, with new ties to many other areas of mathematics and physics."\\\\Keywords:locally compact quantum groups, quantum groupoids, discrete quantum groups, algebraic quantum groups, monoidal category structure, locally compact quantum groupoids, Hopf algebra, von Neumann algebra, duality, unitary corepresentations, weak bialgebras, relative tensor product, multiplicative unitary, corresponding matched pair, faithful weight, multiplicative unitaries, quantum subgroup, involutive homomorphism, face algebra, grouplike elements, partial isometrics, partial isometry, infinitesimal objects, quantum group theory, phase change, monoidal functor, canonical trace, preprint math, dense ideal. See also: http://www-irma.u-strasbg.fr/rubrique83.html

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Copyright@2003 by Walter de Gruyter Gmbh and Co, 10785, Berlin http://www-irma.u-strasbg.fr/rubrique83.html

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