On Braided Lie Structures of Algebras in the Categories of Weak Hopf Bimodules

Abstract

Let H be a weak Hopf algebra. In this paper, it is proved that the monoidal category [Formula: see text] of weak Hopf bimodules studied in Wang [19] is equivalent to the monoidal category [Formula: see text] of weak Yetter–Drinfel'd modules introduced in Böhm [2]. When H has a bijective antipode, a braiding in the category [Formula: see text] is constructed by the braiding on [Formula: see text], generalizing the main result in Schauenburg [14]. Finally, the braided Lie structures of an algebra A in the category [Formula: see text] are investigated, by showing that if A is a sum of two braided commutative subalgebras, then the braided commutator ideal of A is nilpotent.