zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Finite quantum groupoids and their applications. (English) Zbl 1026.17017
Montgomery, Susan (ed.) et al., New directions in Hopf algebras. Cambridge: Cambridge University Press. Math. Sci. Res. Inst. Publ. 43, 211-262 (2002).
The paper surveys some aspects of the theory of weak Hopf algebras or finite quantum groupoids. These are generalisations of finite-dimensional Hopf algebras introduced by {\it G. Böhm, F. Nill} and {\it K. Szlachányi} [J. Algebra 221, 385-438 (1999; Zbl 0949.16037)] in which the coproduct is a multiplicative but non-unital map and the counit is not an algebra map. The authors begin by giving definitions and listing basic examples such as groupoid algebras, quantum transformation groupoids and Temperley-Lieb algebras. Next the theory of integrals in finite quantum groupoids is described including basic facts about weak Hopf modules and the Maschke theorem. The authors then proceed to describe actions and smash products of weak Hopf algebras and the respresentation category of a quantum groupoid. The description of the latter involves discussion of special classes of weak Hopf algebras, in particular quasitriangular and ribbon quantum groupoids, and culminates in the analysis of the relationship between the representation categories of quantum groupoids and modular categories. Next the generalisation of Drinfeld’s twisting construction is made, thus leading to a class of weak Hopf algebras known as dynamical quantum groups and obtained by dynamical twisting of universal enveloping algebras. The final part of the paper is devoted to $C^*$-quantum groupoids. The definition, existence of the Haar measure and the semisimplicity are discussed. Finally the relationship between $C^*$-quantum groupoids and the theory of finite depth subfactors is described. For the entire collection see [Zbl 0990.00022].

17B37Quantum groups and related deformations
46L89Other “noncommutative” mathematics based on $C^*$-algebra theory
16W30Hopf algebras (associative rings and algebras) (MSC2000)
16W35Ring-theoretic aspects of quantum groups (MSC2000)
46L60Applications of selfadjoint operator algebras to physics
Full Text: Link arXiv