On the Structure Theorem for Quasi-Hopf Bimodules
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- 作者：Paolo Saracco
- 关键词：Quasi ; bialgebras ; Structure theorem ; Preantipode ; Right quasi ; Hopf bimodules ; Quasi ; Hopf algebras ; Gauge
- 刊名：Applied Categorical Structures
- 出版年：2017
- 出版时间：February 2017
- 年：2017
- 卷：25
- 期：1
- 页码：3-28
- 全文大小：
- 刊物类别：Mathematics and Statistics
- 刊物主题：Mathematical Logic and Foundations; Theory of Computation; Convex and Discrete Geometry; Geometry;
- 出版者：Springer Netherlands
- ISSN：1572-9095
- 卷排序：25
文摘
The Structure Theorem for Hopf modules states that if a bialgebra A is a Hopf algebra (i.e. it is endowed with a so-called antipode) then every Hopf module M is of the form McoA⊗A, where McoA denotes the space of coinvariant elements in M. Actually, it has been shown that this result characterizes Hopf algebras: A is a Hopf algebra if and only if every Hopf module M can be decomposed in such a way. The main aim of this paper is to extend this characterization to the framework of quasi-bialgebras by introducing the notion of preantipode and by proving a Structure Theorem for quasi-Hopf bimodules. We will also establish the uniqueness of the preantipode and the closure of the family of quasi-bialgebras with preantipode under gauge transformation. Then, we will prove that every Hopf and quasi-Hopf algebra (i.e. a quasi-bialgebra with quasi-antipode) admits a preantipode and we will show how some previous results, as the Structure Theorem for Hopf modules, the Hausser-Nill theorem and the Bulacu-Caenepeel theorem for quasi-Hopf algebras, can be deduced from our Structure Theorem. Furthermore, we will investigate the relationship between the preantipode and the quasi-antipode and we will study a number of cases in which the two notions are equivalent: ordinary bialgebras endowed with trivial reassociator, commutative quasi-bialgebras, finite-dimensional quasi-bialgebras.