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].