Regular multiplier Hopf algebroids II. Integration on and duality of algebraic quantum groupoids
Regular multiplier Hopf algebroids II. Integration on and duality of algebraic quantum groupoids
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发表时间：
2014-03
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通讯作者：
T. Timmermann
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作者：
T. Timmermann
量子群的一个基本特征是，许多量子群是成对的相互对偶的对象，如有限维的霍普夫代数及其子代数，或函数代数的量子化和泊松-李群的泛包络代数。同样的现象被研究了量子群胚在不同的设置。在纯代数体系中，对偶对象的构造由Schauenburg、Kadison和Szlach\'anyi给出，但要求量子广群相对于基是有限的。度量量子群胚的一个复杂的对偶性是由Enock，Lévilles和Vallin在von Neumann代数的背景下发展的。我们提出了一个纯粹的代数对偶理论，没有任何有限性假设，推广货车Daele的对偶理论的乘子Hopf代数和借用的思想，从理论的测量量子群胚。我们的方法是基于乘子的Hopf代数胚最近推出的货车Daele和作者，并在一个新的方法来整合代数量子群胚。主要概念是左和右积分的经常乘子的Hopf代数体，适应准不变的重量的基础上。鉴于这样的积分，我们表明，他们是唯一的重新缩放，承认模块化自同构，以及左，右相关的模块化元素。然后，在没有任何有限性和Frobenius假设的情况下，构造了一个带积分的对偶乘子Hopf代数体，并证明了它的对偶性.
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.