On compact bicrossed products

  • Pierre Fima

    Université Denis Diderot Paris 7, Paris, France
  • Kunal Mukherjee

    Indian Institute of Technology Madras, Chennai, India
  • Issan Patri

    The Institute of Mathematical Sciences, Chennai, India


We make a comprehensive and self-contained study of compact bicrossed products arising from matched pairs of discrete groups and compact groups. We exhibit an automatic regularity property of such a matched pair and describe the representation theory and the fusion rules of the associated bicrossed product . We investigate the relative co-property and the relative co-Haagerup property of the pair comprising of the compact group and the bicrossed product, discuss property and Haagerup property of the discrete dual , and review co-amenability of as well. We distinguish two such non-trivial compact bicrossed products with relative co-property and also provide an infinite family of pairwise non isomorphic non-trivial discrete quantum groups with property , the existence of even one of the latter was unknown. Finally, we examine all the properties mentioned above for the crossed product quantum group given by an action by quantum automorphisms of a discrete group on a compact quantum group, and also establish the permanence of rapid decay and weak amenability and provide several explicit examples.

Cite this article

Pierre Fima, Kunal Mukherjee, Issan Patri, On compact bicrossed products. J. Noncommut. Geom. 11 (2017), no. 4, pp. 1521–1591

DOI 10.4171/JNCG/11-4-10