Weak Tensor Category and Related Generalized Hopf Algebras

Abstract

There are at least two kinds of generalization of Hopf algebra, i.e. pre–Hopf algebra and weak Hopf algebra. Correspondingly, we have two kinds of generalized bialgebras, almost bialgebra and weak bialgebra. Let \({\fancyscript C}\) = (\({\fancyscript C}\) , ⊗, I, a, l, r) be a tensor category. By giving up I, l, r and keeping ⊗, a in \({\fancyscript C}\) , the first author got so–called pre–tensor category \({\fancyscript C}\) = (\({\fancyscript C}\) , ⊗, a) and used it to characterize almost bialgebra and pre–Hopf algebra in Comm. in Algebra, 32(2): 397–441 (2004). Our aim in this paper is to generalize tensor category \({\fancyscript C}\) = (\({\fancyscript C}\), ⊗, I, a, l, r) by weakening the natural isomorphisms l, r, i.e. exchanging the natural isomorphism ll −1 = rr −1 = id into regular natural transformations lll = l, rrr = r with some other conditions and get so–called weak tensor category so as to characterize weak bialgebra and weak Hopf algebra. The relations between these generalized (bialgebras) Hopf algebras and two kinds generalized tensor categories will be described by using of diagrams. Moreover, some related concepts and properties about weak tensor category will be discussed.