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Journal of Noncommutative Geometry

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Volume 11, Issue 4, 2017, pp. 1521–1591
DOI: 10.4171/JNCG/11-4-10

Published online: 2017-12-15

On compact bicrossed products

Pierre Fima[1], Kunal Mukherjee[2] and Issan Patri[3]

(1) Université Denis Diderot Paris 7, Paris, France
(2) Indian Institute of Technology Madras, Chennai, India
(3) 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 $\mathbb G$. We investigate the relative co-property $(T)$ and the relative co-Haagerup property of the pair comprising of the compact group and the bicrossed product, discuss property $(T)$ and Haagerup property of the discrete dual $\widehat{\mathbb G}$, and review co-amenability of $\mathbb G$ as well. We distinguish two such non-trivial compact bicrossed products with relative co-property $(T)$ and also provide an infinite family of pairwise non isomorphic non-trivial discrete quantum groups with property $(T)$, 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.

Keywords: Compact quantum group, bicrossed product, matched pair, property T, Haagerup property

Fima Pierre, Mukherjee Kunal, Patri Issan: On compact bicrossed products. J. Noncommut. Geom. 11 (2017), 1521-1591. doi: 10.4171/JNCG/11-4-10