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 L = ...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 L = （L, ×, I, a, l, r） be a tensor category. By giving up I, l, r and keeping ×, a in L, the first author got so-called pre-tensor category L = （L, ×, 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 L = （L, ×, 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.展开更多
We survey a recent progress on algebraic quantum field theory in connection with subfactor theory. We mainly concentrate on one-dimensional conformal quantum field theory.
基金the Program for New Century Excellent Talents in University(No.04-0522)the National Natural Science Foundation of China(No.10571153)the Natural Science Foundation of Zhejiang Province of China(No.102028)
文摘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 L = （L, ×, I, a, l, r） be a tensor category. By giving up I, l, r and keeping ×, a in L, the first author got so-called pre-tensor category L = （L, ×, 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 L = （L, ×, 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.
文摘We survey a recent progress on algebraic quantum field theory in connection with subfactor theory. We mainly concentrate on one-dimensional conformal quantum field theory.
