Left coideal subalgebras of Nichols algebras
Final Report Abstract
The project was dedicated to left coideal subalgebras in the category of comodules of Nichols algebras of Yetter-Drinfeld modules. The focus was on the study of Nichols algebras over nonabelian groups, with the objective of better understanding their largely unknown structure. Our work led to a new classification of Nichols algebras over groups based on a property of their left coideal subalgebras. More precisely, we determined all Nichols algebras of finite-dimensional Yetter-Drinfeld modules over groups such that all its left coideal subalgebras in the category of N0-graded comodules over the group algebra are generated in degree one as an algebra. Here we confined ourselves to Yetter-Drinfeld modules in which each group-homogeneous component is at most one-dimensional. The result and the proof indicate the complexity of this question. Thereby we obtained explicit construction methods for not primitively generated left coideal subalgebras. Another emphasis was on strengthening a notable known Theorem on the decomposition of a pointed braided Hopf algebra into a tensor product of a left coideal subalgebra and a quotient coalgebra right module. Commonly one considers pointed braided Hopf algebras A in the category of Yetter-Drinfeld modules over a Hopf algebra H. In this case the braided Hopf algebra and also many interesting left coideal subalgebras K are in particular comodules and it is desirable that the decomposition is compatible with the comodule structure. We proved the compatibility under the assumption that the underlying ordinary Hopf algebra is cosemisimple and under some weak additional assumptions. Thereby we proved that Hopf modules which are left A-comodules and right K-modules in the category of left H-comodules (satisfying a compatibility condition) are free as K-modules in the category of left H-comodules.
Publications
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A decomposition Theorem for pointed braided Hopf algebras. Communications in Algebra, 51(9), 3904-3915.
Heckenberger, Istvan & Schäfer, Katharina
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Left coideal subalgebras of Nichols algebras. Contemporary Mathematics, 97-149. American Mathematical Society.
Heckenberger, Istvan & Schäfer, Katharina
