DOUBLE CROSSED PRODUCTS OF LOCALLY COMPACT QUANTUM GROUPS

Abstract

For a matched pair of locally compact quantum groups, we construct the double crossed product as a locally compact quantum group. This construction generalizes Drinfeld’s quantum double construction. We study the modular theory and the $\mathrm{C}^*$-algebraic properties of these double crossed products, as well as several links between double crossed products and bicrossed products. In an appendix, we study the Radon–Nikodym derivative of a weight under a quantum group action (following Yamanouchi) and obtain, as a corollary, a new characterization of closed quantum subgroups. AMS 2000 Mathematics subject classification: Primary 46L89. Secondary 46L65