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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 is determined by the commutativity and cocommutativity of , where is a weak Hopf algebra with a bijective antipode. [ABSTRACT FROM AUTHOR]

In this article we first study an equivariant cyclic cohomology for weak H-module agebras over a weak Hopf algebra H with a bijective antipode. Then we define an equivariant K-theory for weak quantum Yetter–Drinfeld algebras over H and establish a generalized Connes' pairing between the equivariant cyclic cohomology and the equivariant K-theory. As an application we consider our theory for groupoids. [ABSTRACT FROM AUTHOR]

Let A be a regular multiplier Hopf algebra, and let Aut(A) denote the set of all isomorphisms α from A to itself that are algebra maps satisfying (Δ ○ α)(a) = (α ⊗ α) ○ Δ(a) for all a ∈ A. Let G be a certain crossed product group Aut(A) × Aut(A). The main purpose of this article is to provide a class of new braided T-categories in the sense of Turaev [\citealp9]. For this, we introduce a class of new categories A 𝒴𝒟 A (α, β) of (α, β)-Yetter–Drinfel'd modules with α, β ∈Aut(A), and we show that the category ℳ𝒴𝒟(A) = { A 𝒴𝒟 A (α, β)}(α, β)∈G becomes a braided T-category over G, generalizing the main constructions by Panaite and Staic [6]. [ABSTRACT FROM PUBLISHER]

Let B and H be weak Hopf algebras with bijective antipodes SB and SH, respectively. Based on a compatible weak Hopf dual pairing (B, H, σ), we construct a generalized Drinfeld quantum double (B, H) which is a weak T-coalgebra over a twisted semi-direct square of groups. In particular, when B and H are finite dimensional and the above pairing map σ is nondegenerate, (B, H) admits a nontrivial quasitriangular structure. Some explicit examples are given as an application of our theory. [ABSTRACT FROM AUTHOR]

The paper is devoted to studying certain families of inner deformations of entwined modules and investigating simple entwined modules. Our main results can be applied to the study of the inner deformation of all kinds of Hopf modules and the irreducible representations of a wide class of Hopf algebras constructed by smash products. As an explicit application, we consider the case of quantum Yetter-Drinfel'd modules. [ABSTRACT FROM AUTHOR]

Let < A , B > be a pairing of two regular multiplier Hopf algebras A and B . One method of constructing the Drinfel'd double &Dscr; = A ⋈ B cop is by the use of an invertible twist map R : B &CircleTimes; A → A &CircleTimes; B defining an associative product on A &CircleTimes; B . In Delvaux (2003) and Drabant and Van Daele (2001), the authors construct R by , where R 2 and is only related to the module actions between A and B . Another way is given in Delvaux and Van Daele (2004a) in which the authors also just consider the module actions and then construct the Drinfel'd double &Dscr; as an algebra of operators on the vector space B &CircleTimes; A . In this article we will give two different points of view of constructing the Drinfel'd double &Dscr; for multiplier Hopf algebras. The first is that the Drinfel'd double &Dscr; associated to the pairing < A , B > is constructed by using not only the module actions but also the comodule coactions, i.e., the Drinfel'd double &Dscr; is given in the framework of a special twisted tensor product algebra structure A &dotsquare; A op &CircleTimes; A B . The second is as follows: If P is a multiplier Hopf algebra and a reduced ( A , B )-bicomodule algebra (an A -Long module algebra), then we present a twisting construction of the product of P via the coactions of A and B on P , and we show that the Drinfel'd double &Dscr; is isomorphic as a multiplier Hopf algebra to the opposite twisting of ( A op , cop &CircleTimes; B op , cop ). As an application of our theory, we consider the case of group-cograded multiplier Hopf algebras and the case of Hopf group-coalgebras. [ABSTRACT FROM AUTHOR]

Describes the nature of Hopf algebras. Structure of Lie coalgebra; Category of left H-modules; Definition of braided Lie algebra.

Examines the quasitriangularity structure of twisted smash coproduct of Hopf algebras. Role of Hopf algebras in quantum groups; Derivation of comultiplications and counit; Theorems applied.

We find a natural compatible condition between the Rota-Baxter operator and Turaev's (Hopf) group-(co)algebras, which leads to the concept of Rota-Baxter Turaev's (Hopf) group-(co)algebra. Two characterizations of Rota-Baxter Turaev's group-algebras (abbr. T-algebras) are obtained: one by Atkinson factorization and the other by T-quasi-idempotent elements. The relations among some related Turaev's group algebraic structures (such as (tri)dendriform T-algebras, Zinbiel T-algebras, pre-Lie T-algebras, Lie T-algebras) are discussed, and some concrete examples from the algebras of dimensions 2,3 and 4 are given. At last we prove that Rota-Baxter Poisson T-algebras can produce pre-Poisson T-algebras and Poisson T-algebras can be obtained from pre-Poisson T-algebras. [ABSTRACT FROM AUTHOR]