On The Classification Of Finite Quasi-quantum Groups

In this dissertation, we will study the classification theory and structure theory of finite-dimensional pointed Majid algebras, and Nichols algebras of diagonal type with finite root systems in some important twisted Yetter-Drinfeld categories. We provide a general method to resolve the abelian 3-cocycles of finite groups, and then with ap-plication of guage transformation of tensor categories, we can reduce the classification problem of Nichols algebras of diagonal type in twisted Yetter-Drinfeld categories, to those in ordinary Yetter-Drinfeld categories. Combine Heickenberger’s classification of arithmetic root systems, we obtain the classification of Nichols algebras of diagonal type with finite root systems in twisted Yetter-Drinfeld categories over finite groups. In par-ticular, we get the classification of the finite-dimensional Nichols algebras of diagonal type in such categories. We prove that all finite-dimensional pointed Majid algebras of diagonal type are generated by group-like elements and skew-primitive elements, i.e., we give a partial positive answer to generalized Andruskiewitsch-Schneider conjecture. Combine this result and the classification of Nichols algebras of diagonal type, we ob-tain the classification of finite-dimensional connected coradically graded pointed Majid algebras of diagonal type. This dissertation consists of five chapters.In the first chapter, we introduce the history and development about quasi-quantum groups. We mainly introduce some current results on the classification theory and clas-sification methods of finite quasi-quantum groups. At last we present the main results of this dissertation.In the second chapter, we introduce the definitions and some basic facts about quasi-quantum groups, tensor categories, Nichols algebras, arithmetic root systems and Weyl groupoids. Some of our recent results about pointed Majid algebras are also introduced, such as the explicit formulae of quasi-version of Majid’s bosonization.In the third chapter, we study the Nichols algebras of diagonal type in twisted Yetter-Drinfeld category KGKGyDΦ, and provide a classification of Nichols algebras of di-agonal type with finite root systems in KGKGyDΦ. Note that the associativity of KGKGyDΦ is determined by 3-cocycle Φ over G. We first prove that if a supply group of a Nichols algebra of diagonal type in KGKGyDΦ then G is an abelian group and Φ must be an abelian 3-cocycle over G. This equals to say that any Nichols algebras of diagonal type can be realized in KGKGyDΦ, with G an abelian group and Φ an abelian 3-cocycle over G. Next we study the abelian 3-cocycles over abelian groups. We found that such 3-cocycles can be resolved through a bigger abelian group. Then we can get the information of the Nichols algebras of diagonal type in KGKGyDΦ,through the Nichols algebras of diagonal type in ordinary Yetter-Drinfeld categories KGKGyD over some bigger abelian group G. In this way, we get the classification of Nichols algebras of diagonal type with finite root systems in KGKGyDΦ.By considering the nilpotent index of root vector associated to each positive root, we get the classification of finite-dimensional Nichols algebras of diagonal type in KGKGyDΦ.In the fourth chapter, we classify all the finite-dimensional connected coradically graded pointed Majid algebras of diagonal type. A Majid algebra is called connected if it’s Gabriel quiver is connected. The general classification of finite-dimensional corad-ically graded pointed Majid algebras can be reduced to the connected case. In order to give a classification of finite-dimensional pointed Majid algebras, we must deal with the generalized Andruskiewitsch-Schneider conjecture. As one of the main results of this dissertation, we prove the conjecture conditional, i.e., any finite-dimensional pointed Majid algebra of diagonal type is generated by group-like elements and skew-primitive elements. With this result, we can reduce the classification problem of finite-dimensional coradically graded pointed Majid algebras of diagonal type to the classification of finite-dimensional Nichols algebras of diagonal type. We also provide some structure theorems of finite-dimensional pointed Majid algebras of diagonal type.In the fifth chapter, the pointed Majid algebras of Cartan type and standard type are studied. We prove that for every finite Cartan matrix, we can attach it infinity many finite-dimensional pointed Majid algebras of Cartan type. At the same time, we provide a general way to construct finite-dimensional pointed Majid algebras of Cartan type from finite Cartan Matrices. We also study the pointed Majid algebras of standard type, and provide a more explicit classification and structure theory of this kind of Majid algebras. At last, we give a large class of examples of rank 2 pointed Majid algebras of diagonal type with finite many PBW generators, and provide a list of rank 2 finite-dimensional pointed Majid algebras of standard type.