Pseudotriangular weak Hopf algebras

In this paper, in looking for the weak Hopf algebraic counterpart of pseudosymmetric braidings, we introduce the concept of a pseudotriangular weak Hopf algebra, which is a quasitriangular weak Hopf algebra satisfying an extra condition. Then, we investigate the question, when a quasitriangular weak Hopf algebra is pseudotriangular. As an application, we study a special class of pseudotriangular weak Hopf algebras, under the name almost-triangular weak Hopf algebras and list some nontrivial examples. Finally, in order to construct more examples of pseudotriangular weak Hopf algebras, we show that the pseudosymmetry of the Yetter-Drinfeld category (HYD)-Y-H is determined by the commutativity and cocommutativity of H, where H is a weak Hopf algebra with a bijective antipode.