Generalized Biproduct

다국어 초록 (Multilingual Abstract)

The concept of a Hopf algebra was developed by algebraic topolotists abstracting the work of Hopf [6] on manifolds admitting a product (such as Lie groups). A basic reference is the famous article [7] by Milnor and Moore. Hopf algebras also came up in the representation theory of Lie groups and algebraic groups (see [8], [9], [10], [11]). For abstract Hopf algebras, we refer to Abe's and Sweedler's monographs [4], [8].
The smash product algebra and the smash coproduct coalgebra are well konwn in the context of Hopf algebra [2], [4] and these notions can be viewed as being motivated by semi-direct product construction in the theory of groups and in the theory of affine group schemes, respectively, In 1985, D.E. Radfored defined biproduct using the smash product and Molnar's smash coproduct. In this thesis we define the generalized smash coproduct coalgebra and define the generalized biproduct using the generalized smash product and generalized smash coproduct.
In Section2, we will define some definitions and give some examples. Let H be a bialgebra over a field k and let B be a left H-module algebra and be also a left H-comodule coalgebra. Let D be a bialgebra and a left H-comodule algebra and be also a left H-module coalgebra. In Section 3, we will find necessary and sufficient conditions for the generalized smash product algebra structure and the generalized smash coproduct coalgebra structure on Bⓧ_(k)D to afford B×^(L)_(H)D a bialgegra stucture. This particular situation has been considered in [3] when the bialgebra D=H since H is a left H-comodule algebra via △_(H) and H is a left H-module coalgebra via m_(H)

목차 (Table of Contents)

• CONTENTS
• I. Introduction = 1
• II. Preliminaries = 2
• III. Generalized Biproduct = 6
• References = 26