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.

Structure Theorems for Bicomodule Algebras Over Quasi-Hopf Algebras, Weak Hopf Algebras, and Braided Hopf Algebras

“…It was proved in [30] that there exists an equivalence of categories between H H M H H and the category of right-right Yetter-Drinfeld modules over H. We will need the left-handed analogue of this result, whose proof is analogous to the one in [30] and is left to the reader:…”

“…Let H be a weak Hopf algebra with bijective antipode. It was proved in [30] that there exists an equivalence of categories between H H M H H and the category of right-right Yetter-Drinfeld modules over H. We will need the left-handed analogue of this result, whose proof is analogous to the one in [30] and is left to the reader: Proposition 2.5 Let H be a weak Hopf algebra with bijective antipode. (i) Let V ∈ H H YD, with H-action denoted by ⊲ and H-coaction V → H ⊗ V , v → v (−1) ⊗ v (0) .…”

Twisting theory for weak Hopf algebras

“…By [11], the braiding is determined by ψ(hv ⊗ wg) = hw ⊗ vg, with h, g ∈ H, v ∈ V left coinvariant and w ∈ W right coinvariant.…”

“…Then (4) , v (1) g (3) )σ −1 (v (2) , g (4) )), and (1) ) ⊗ Hσ σ(w (−1) , g (1) )w (0) g (2) σ −1 (w (1) , g (3) )) = ψ(σ(h (1) , 1 (1) )σ −1 (h (3) , v (2) )σ(h (2) (1) 1 (2) , w (−1) g (2) (1) )h (2) (2) (3) )σ(w (−2) , g (1) )σ −1 (1 (3) , g (3) )) (11) (4) , v (2) )σ(h (1) , w (−1) g (2) )h (2) (1) , g (4) )σ(w (−2) , g (1) ) (2) , g (4) )), where we have used the properties of the left coinvariant and the right coinvariant of v and w. This finishes the proof. …”

The Generalized C.M.Z.-Theorem and a Drinfel'd Double Construction for WT-Coalgebras and Graded Quantum Groupoids

“…In the articles Zunino (2004a,b) the author provided an analogue of an ordinary Drinfel'd double construction and introduced the notion of a YetterDrinfel'd module for T -coalgebras of finite type, which were recently generalized to the setting of WT-coalgebras in Van Daele and and in Wang and Zhu (2008), respectively. In particular, Wang (2004) proved that the C.M.Z.-theorem holds for T -coalgebras and that the Drinfel'd double construction for T -coalgebras appears as a type of -twisted smash product.…”