【作者】 刘公祥；

【导师】 李方；

【作者基本信息】 浙江大学 ， 基础数学， 2005， 博士

【摘要】 本论文的主要目的是分类有限维的Hopf代数，特别地去分类有限维的基本Hopf代数。我们的思想是通过其表示型来分类他们，我们的方法主要依赖于有限维代数的表示理论。 为了分类有限维的Hopf代数，我们给出了以下四个步骤： （1）给出一个有效的方法来决定基本Hopf代数的表示型； （2）通过表示型来分类有限维基本Hopf代数； （3）确定一个Hopf代数什么时候会Morita等价于一个基本Hopf代数； （4）寻找新的途径来将（2）的结果推广到一般的有限维Hopf代数。 为了解决步骤（1），我们为每一个基本Hopf代数H配备了一个被称为表示型数的数n_H并且证明（ⅰ）H是有限型当且仅当n_H=0或n_H=1；（ⅱ）如果H是Tame型，则n_H=2；（ⅲ）如果n_H≥3，则H是Wild型。 对于（2），目前，我们已经给出了有限型的完整分类。具体地讲，它们共分三类：①如果H是半单的，则H同构与一个群代数的对偶；②如果H是非半单的并且基础域的特征是0的话，则H同构一个所谓Andruskiewitsch-Schneider代数与一个群代数交差积的对偶；③如果H是非半单的并且基础域的特征不是0的话，则H同构于某个特定代数与一个群代数交差积的对偶。对于Tame型的Basic Hopf代数，我们可以给出的是根分次情形的结构定理。我们将看到根分次的情形至多只有五类。我们还给出了一些关于TameHopf代数的例子。 广义路（余）代数为我们解决步骤（3）（4）提供了一种可能。我们首先研究了所谓的广义路余代数的同构问题，证明了两个广义路余代数k（△，C）≌k（△′，D）当且仅当存在quiver的同构ψ：△→△′使得S_i≌T_ψ（i）对任意i∈△₀。我们还给出了广义路（余）代数的Gabriel’s定理。关于一个广义路余代数上什么时候具有Hopf代数结构的问题也被解决。更多还原

【Abstract】 The main aim of this paper is to classify finite dimensional Hopf algebras, especially basic Hopf algebras. Our idea is to classify them through their representation type and our methods relay heavily on the representation theory of finite dimensional algebras.In order to do so, we give four programs to classify finite dimensional Hopf algebras as follows.(1) Give an effective way to determine the representation type of a finite dimensional basic Hopf algebra;(2) Classify finite dimensional basic Hopf algebras through their representation type;(3) Determine that when a finite dimensional Hopf algebra is Morita equivalent to a finite dimensional basic Hopf algebra;(4) Find some new ways to generalize the conclusions in (2) to general finite dimensional Hopf algebras.In order to resolve program (1), we attache every finite dimensional basic Hopf algebra H a number nh which is called representation type number of H and proved that (i) H is of finite representation type if and only if nH = 0 or nH = 1; (ii) H is tame then nh = 2 and (iii) H is wild if nH≥ 3.For program (2), we can classify finite dimensional basic Hopf algebras of finite representation type completely now. Explicitly, they are consist of three classes: (i) If H is semisimple, then H (?) k(G)* for some finite group; (ii) If H is not semisimple and the characteristic of k is zero, then H is isomorphic the dual of the cross product between one so called Andruskiewitsch-Schneider algebra and a group algebra; (iii) If H is not semisimple and the characteristic of k is not zero, then H is isomorphic the dual of the cross product between one special algebra and a group algebra. We also can give the structure theorem for finite dimensional basic Hopf algebra of tame type in the radical graded case. We can see that in this case they are consist of five classes at most. More examples about tame Hopf algebras are also given in this paper.Generalized path (co) algebras give us one possibility to solve programs (3) (4). We study so called isomorphism problem for generalized path coalgebras at first and prove that two normal generalized path coalgebras k(△,C) (?)(△’,D) as coalgebras if and only if there is an isomorphism of quivers (?) : △ →△’ such that Si (?) T(?)(i) as coalgebras for i ∈ △0. The Gabriel’s Theorem for generalized path (co)algebras are also given in this paper. The problem of when there is a Hopf structure on generalized path coalgebra is settled.更多还原