Classification Of Finite Dimensional Basic Hopf Algebras And Related Topics

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.