A Generalization for -Cocycles

We will give generalized definitions called type II -cocycles and weak quasi-bialgebra and also show properties of type II -cocycles and some results about weak quasi-bialgebras, for instance, construct a new structure of tensor product algebra over a module algebra on weak quasi-bialgebras. 1. Introduction We will introduce a new generalized definition called type II -cocycle; the namely, relax the invertible condition of associator in -cocycle up to adding an associator satisfies all forms in definition of -cocycle together with the original associator, and both need not to be invertible for each other; then we give examples to illustrate it clearly. Majid have shown many results about -cocycle in [1], and we obtain some results including cohomologous concept through this new definition, main properties of type II -cocycles, and its simple classification. It is well known that quasi-bialgebras and quasi-Hopf algebras play important roles in quantum group theory, and these concepts were introduced by Drinfel’d in [2] who relaxed the coassociative law of up to conjugation. In this paper, we will show a new definition called weak quasi-bialgebra, a generalization of quasi-bialgebras, and there are simple examples to illustrate. Authors show results for weak quasi-bialgebras in place of quasi-bialgebras (cf. [1, 3]), including that there exists an algebra structure on , a generalization of the algebra product in [3], where is a weak quasi-bialgebra and is a -module algebra. We follow all the notation and conventions in [1], throughout the paper. In the following, will be a fixed field throughout, and all algebras, coalgebras, vector spaces, and so forth are over automatically unless specified. We recall definitions as follows. Definition 1.1. For any bialgebra, if there is an invertible element such that where integers and are max even number and max odd number, respectively, in , we call an -cocycle. If ( ), then the cocycle is counital. We define , , , and . Definition 1.2. Let be a algebra with unit and homomorphisms , . If there exists a counital 3-cocycle rendering that, for all , Then is called a quasi-bialgebra together with coproduct and counit , and call an associator. We will denote the tensor components of with big and small letters, respectively, for instance, where is the ith factor. Definition 1.3. Let be a quasi-bialgebra and a vector space. If has a multiplication and the unit obeying that for any and , where is the module structure map of , then say is a left module algebra. In bialgebras, the composition of any two algebra homomorphisms