Pairing and duality of algebraic quantum groupoids

Algebraic quantum groupoids have been developed by two of the authors this note (AVD and SHW) in a series papers [A. Van Daele S. Wang, Weak multiplier Hopf algebras. Preliminaries, motivation basic examples, Banach Center Publ. 98 (2012) 367–415; A. algebras I. The main theory, J. Reine Angew. Math. 705 (2015) 155–209; II. source target algebras, preprint (2014), arXiv: 1403.7906v2 [math.RA]; III. Integrals duality, (2017), 1701.04951 [math.RA]], see also Daele, groupoids. An example, 1702.04903 [math.RA]]. By an algebraic groupoid, we understand regular weak algebra with enough integrals. Regular algebroids are obtained (TT AVD) [T. Timmermann algebroids. Basic theory Commun. Algebra 46 (2017) 1926–1958]. Integral duality for those studied one author here (TT) Timmermann, Integration on groupoids, Int. 27 (2016) 1650014, 1507.00660 [QA]; T. On Adv. 309 692–746, 1605.06384 [math.QA]]. In these papers, term groupoid is used algebroid single faithful integral. Finally, again us investigated relation between Multiplier arising from 106 73–110]. paper, results that dual admits same type. [math.QA]], result nature integral can be (see [math.QA]]). That however not obvious way to obtain result. It more difficult less natural than direct followed We will discuss statement further paper. Nevertheless, it interesting investigate approaches greater detail. This what do build intimate as now add presence seems done best framework pairs. fact general objects coming convinced material present paper provide deeper understanding both within well generally feel appropriate include some historical comments development theories.