Regular multiplier Hopf algebroids II. Integration on and duality of   algebraic quantum groupoids
  A fundamental feature of quantum groups is that many come in pairs of mutually dual objects, like finite-dimensional Hopf algebras and their duals, or quantisations of function algebras and of universal enveloping algebras of Poisson-Lie groups. The same phenomenon was studied for quantum groupoids in various settings. In the purely algebraic setup, the construction of a dual object was given by Schauenburg and by Kadison and Szlach\'anyi, but required the quantum groupoid to be finite with respect to the base. A sophisticated duality for measured quantum groupoids was developed by Enock, Lesieur and Vallin in the setting of von Neumann algebras.   We propose a purely algebraic duality theory without any finiteness assumptions, generalising Van Daele's duality theory of multiplier Hopf algebras and borrowing ideas from the theory of measured quantum groupoids. Our approach is based on the multiplier Hopf algebroids recently introduced by Van Daele and the author, and on a new approach to integration on algebraic quantum groupoids. The main concept are left and right integrals on regular multiplier Hopf algebroids that are adapted to quasi-invariant weights on the basis. Given such integrals, we show that they are unique up to rescaling, admit modular automorphisms, and that left and right ones are related by modular elements. Then, we construct, without any finiteness or Frobenius assumption, a dual multiplier Hopf algebroid with integrals and prove biduality. 
Thomas Timmermann
math.QA math.OA