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On the classification of finite-dimensional pointed Hopf algebras. (English) Zbl 1208.16028

The authors give the classification of finite-dimensional complex Hopf algebras which are pointed, that is, all of whose irreducible comodules are one-dimensional, and whose group of group-like elements is Abelian with prime divisors of the order bigger than 7. This result is one of the few general classification results for Hopf algebras known so far. It can be read as an axiomatic description of generalized small quantum groups. The Hopf algebras are described by generators and relations and it is shown that they are the small quantum groups discovered by Lusztig and variations of them.
The main result says that a pointed finite-dimensional complex Hopf algebra with Abelian group \(\Gamma\) of group-like elements having no prime divisors \(\leq 7\) is necessarily isomorphic to one of the form \(u(\mathcal D,\lambda,\mu)\), as described in Section 4.2 of the paper, where \(\mathcal D\) is a datum of finite Cartan type for the group \(\Gamma\), and \(\lambda\) and \(\mu\) are finite families of free parameters in \(k\). In order to accomplish the classification of the Hopf algebras under consideration, the authors apply the so called Lifting Method, developed in their previous joint work; [see J. Algebra 209, No. 2, 658-691 (1998; Zbl 0919.16027)], which works for Hopf algebras whose coradical is a Hopf subalgebra. Other essential ingredients in the proof of the main result are the results of I. Heckenberger on Nichols algebras of diagonal type [Invent. Math. 164, No. 1, 175-188 (2006; Zbl 1174.17011)], which use V. K. Kharchenko’s theory [Algebra Logika 38, No. 4, 476-507 (1999); translation in Algebra Logic 38, No. 4, 259-276 (1999; Zbl 0936.16034)] of PBW-bases in braided Hopf algebras of diagonal type.
Several consequences of the main result are shown, like a version of Cauchy’s theorem from finite group theory for the Hopf algebras \(u(\mathcal D,\lambda,\mu)\), and further support to the open conjecture that a pointed Hopf algebra is always generated by group-like and skew-primitive elements.

MSC:

16T05 Hopf algebras and their applications
17B37 Quantum groups (quantized enveloping algebras) and related deformations
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References:

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