Lifting of Nichols Algebras

Nichols algebras are a fundamental building block of pointed Hopf algebras. Part of the classification program of finite-dimensional pointed Hopf algebras with the lifting method of Andruskiewitsch and Schneider is the determination of the liftings, i.e., all possible deformations of a given Nichols algebra. The classification was carried out in this way in several special cases when the appearing Nichols algebras are of Cartan type. We compute explicitly the liftings of some Nichols algebras not of Cartan type or of Cartan type without restrictions on the group order. These build up new classes of finite-dimensional pointed Hopf algebras. Furthermore, we give a necessary and sufficient PBW basis criterion for a class of pointed Hopf algebras and present them in terms of generators and relations. These Hopf algebras can be seen as generalized Quantum groups.

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born 1980 in Weiden, Germany,studied mathematics at the Universities of Munich and Strasbourg.Diploma in 2005 and Doctoral Degree in 2009. Since 2006 researchassistant at the University of Munich. 2006-2008 scholarship ofthe University of Bavaria. Since 2007 lecturer at the Universityof Applied Sciences of Munich.