Symmetric centres of braided monoidal categories

This paper introduces the concept of ‘symmetric centres’ of braided monoidal categories. LetH be a Hopf algebra with bijective antipode over a fieldk. We address the symmetric centre of the Yetter-Drinfel’d module category:[graphic not available: see fulltext] and show that a left Yetter-Drinfel’d moduleM belongs to the symmetric centre of[graphic not available: see fulltext] and only ifM is trivial. We also study the symmetric centres of categories of representations of quasitriangular Hopf algebras and give a sufficient and necessary condition for the braid of,Hℳ to induce the braid of[graphic not available: see fulltext], or equivalently, the braid of[graphic not available: see fulltext], whereA is a quantum commutativeH-module algebra