On some Nichols algebras with finite Cartan graph

We apply the reflection theory of Yetter-Drinfeld modules developed by István Heckenberger and Hans-Jürgen Schneider to study the Nichols algebra of a certain three-dimensional Yetter-Drinfeld module arising from rigid braided vector spaces. This Yetter-Drinfeld module decomposes into a two- and a one-dimensional Yetter-Drinfeld module. As a first main result, the existence of the Cartan graph associated with the Nichols algebra of Yetter-Drinfeld module under an additional assumption has been studied and a classification of occurring Cartan graphs was obtained. Moreover, the Cartan graphs that are 'finite' were identified. As a consequence, new examples of finite-dimesional Nichols algebras are obtained whose dimensions could be stated explicitly.