Generalized (anti) Yetter-Drinfeld modules as components of a braided T-category

Abstract

If H is a Hopf algebra with bijective antipode and α, β ∈ Aut Hopf (H), we introduce a category H YD H (α, β), generalizing both Yetter-Drinfeld modules and anti-Yetter-Drinfeld modules. We construct a braided T-category YD(H) having all the categories H YD H (α, β) as components, which if H is finite dimensional coincides with the representations of a certain quasitriangular T-coalgebra DT (H) that we construct. We also prove that if (α, β) admits a so-called pair in involution, then H YD H (α, β) is isomorphic to the category of usual YetterDrinfeld modules H YD H .

Constructing New Braided T-Categories over Weak Hopf Algebras

“…Then the category H WYD H (α, β) is just the category of (α, β)-Yetter-Drinfeld modules studied in [14]. …”

“…The aim of this paper is to generalize the above constructions by Panaite and Staic [14], replacing their Hopf algebra H by a weak Hopf algebra in the sense of Böhm et al [1]. This provides further examples of Turaev's T-categories.…”

“…For this, in [14], the authors generalized left-right YetterDrinfeld modules H YD H over a Hopf algebra H with bijective antipode to (α, β)-Yetter-Drinfeld modules, for any Hopf algebra automorphisms α and β of H. They constructed a braided T-category YD(H) over a certain crossed product group G := Aut Hopf (H) × Aut Hopf (H). The component of YD(H) at (α, β) ∈ G is the category of (α, β)-Yetter-Drinfeld modules.…”

“…As the main body of this paper, we construct in Section 4 a class of new braided T-categories WYD(H) in the sense of Turaev [17], having all the weak (α, β)-YetterDrinfeld modules categories H WYD H (α, β) as components (see Theorem 4.6), generalizing thus the main result from Panaite and Staic [14].…”

Another construction of the braided T-category

“…For this, Panaite and Staic (see [6]) and Zunino (see [13]) gave some interesting constructions in Hopf algebra case. Following their motivated idea, Yang et al (see [9,10,12]) generalized their work to multiplier Hopf algebra and weak Hopf algebra cases, also the authors get some new classes of braided T -categories.…”