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Finite-dimensional pointed Hopf algebras with alternating groups are trivial. (English) Zbl 1234.16019

The authors obtain further results on the classification of pointed Hopf algebras over the field \(\mathbb C\) of complex numbers. This is done according to the general method (so-called Lifting Method) developed by the first-named author and H.-J. Schneider [Math. Sci. Res. Inst. Publ. 43, 1-68 (2002; Zbl 1011.16025)]. In order to classify pointed Hopf algebras \(H\) with a finite group \(G\) of group-like elements, a key step is to determine whether the dimension of the Nichols algebra associated to a Yetter-Drinfeld module over \(G\) is finite.
The main result of the paper under review states that Nichols algebras over the alternating group \(A_n\), \(n\geq 5\), are infinite dimensional. This implies that every finite dimensional pointed Hopf algebra with group of group-likes isomorphic to \(A_n\), \(n\geq 5\), is isomorphic to the group algebra of \(A_n\). On the other hand, it is shown that Nichols algebras over the symmetric group \(S_n\), \(n\geq 5\), are infinite dimensional, except maybe those quadratic algebras related to transpositions appearing in the work of S. Fomin and A. N. Kirillov [Prog. Math. 172, 147-182 (1999; Zbl 0940.05070)] and one further class in the case of \(S_5\). It is also shown that for any simple rack \(X\) arising from a symmetric group, except for those in a small list, and for any \(2\)-cocycle on \(X\) the dimension of the corresponding Nichols algebra is infinite. This improves previous results for the symmetric group obtained by N. Andruskiewitsch and F. Fantino [J. Math. Phys. 48, No. 3, 033502 (2007; Zbl 1112.16035)] and N. Andruskiewitsch, F. Fantino and S. Zhang [Manuscr. Math. 128, No. 3, 359-371 (2009; Zbl 1169.16022)].

MSC:

16T05 Hopf algebras and their applications
17B37 Quantum groups (quantized enveloping algebras) and related deformations
16S34 Group rings

Software:

RiG; GAP
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

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