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Representation rings of pointed Hopf algebras of finite type (R-4169)
The computation of the representation ring (or the Green ring) of a Hopf algebra (or a group) is one of the most difficult topics in the Representation Theory of Hopf algebras (or groups). However, the resent research progress made by UHasselt, UAntwerpen and Yangzhou University sheds light on the development on this topic. This proposal forms one part of the ongoing joint research project between the Algebra group of UHasselt and the algebra group of Yangzhou University. The proposed project aims to study the representation rings of finite dimensional pointed Hopf algebras of finite type in the next two years. In the first year of his Ph.D. research, the candidate has successfully computed the representation rings of pointed Hopf algebras of rank one, an important class of pointed Hopf algebras of finite type. The results in his preprint  generalize the results of Li and Zhang  and the results of Chen, Van Oystaeyen and Zhang . In this proposal, the candidate will compute the representations rings of the other classes of pointed Hopf algebras of finite type, and study the properties of the representation rings such as the semisimplicity, commutativity, semiprime, etc. It is expected that the ring properties of the representation ring of a Hopf algebra H will reflect the structure of the monoidal category of representation over H. The results from this work will make great contribution to the knowledge of the Representation Theory of Hopf algebras and quantum groups.  Z. Wang, Representation rings of pointed Hopf algebras of rank one of nilpotent type, preprint.  L. Li and Y. Zhang, The Green rings of the generalized Taft algebras, to appear in Contemporary Mathematics, AMS.  H.Chen, F. Van Oystaeyen and Y.Zhang, The Green rings of Taft algebras, to appear in Proc. AMS.
Date:2 Feb 2013 → 31 Jul 2014