Abstracts Category : Other

On finite tensor categories and quasi-quantum groups

by Yuping Yang

This dissertation is devoted to the classification theory and structure theory of finite-dimensional basic quasi-Hopf algebras and finite pointed tensor categories. Our main results are fourfold, and can be summarized as follows.Firstly, we study the Nichols algebras of twisted Yetter-Drinfeld modules over an abstract finite abelian group G, including those of nondiagonal type. The method is to associate each Nichols algebra of a twisted Yetter-Drinfeld module to an ordinary Nichols algebra, and then apply Heckenberger's classification result on arithmetical root systems. As result, all the finite-dimensional Nichols algebras of simple twisted Yetter-Drinfeld modules are determined. When the order of G is odd, the finite-dimensional Nichols algebras of twisted Yetter-Drinfeld modules over G are classified.Secondly, we investigate the construction of finite pointed tensor categories over an algebraical closed field of characteristic zero. We give a partial positive answer to a conjecture of Etingof, Glaki, Nikshych and ostrich, i.e. we prove that each finite pointed tensor category with abelian Grothendieck group of odd order is tensor generated by objects of length 2. This amounts to saying that any finite-dimensional pointed Majid algebras over an abelian group of odd order is generated by group-like elements and skew-primitive elements.Thirdly, we provide a conceptional classification of odd-dimensional coradically graded pointed Majid algebras over abelian groups. This is also equivalent to a classification of coradically graded pointed Majid algebras over abelian groups of odd orders.Finally, we construct classes finite-dimensional quasi-Hopf algebras from finite abelian groups and braided vector spaces of Cartan type. These quasi-Hopf algebras are nonradically graded in general, and lead to several interesting results such as the classification and structure theories of basic quasi-Hopf algebras with certain conditions. As special cases of quasi-Hopf algebras of Cartan type, the small quasi-quantum groups are introduced and studied. Many other new examples of finite-dimensional genuine quasi-Hopf algebras are also provided, and consequently the associated finite tensor categories, are thus obtained. Deze thesis is gewijd aan de classificatietheorie en structuurtheorie van eindig dimensionale basic quasi-Hopf algebras en eindige gepunte tensorcategorien. De hoofdresultaten zijn vierledig en kunnen samengevat worden als volgt. Eerst en vooral classificeren we Nichols algebras van getwiste Yetter-Drinfeld modulen over een eindige abelse groep G, inclusief deze van het niet-diagonale type. De classificatiemethode bestaat eruit om aan elke Nichols algebra van een getwist YetterDrinfeld moduul, een gewone Nichols algebra te koppelen, vervolgens kan men Heckenbergers classificatieresultaat van aritmetische wortelsystemen gebruiken. Bijgevolg zijn alle eindig dimensionale Nichols algebras van enkelvoudige getwiste Yetter-Drinfeldmodulen bepaald. Indien de orde van de groep G oneven is, zijn deAdvisors/Committee Members: Zhang, Yinhuo, Huang, Hua-Lin.