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Journal of Zhejiang University SCIENCE A 2006 Vol.7 No.6 P.1077-1083

http://doi.org/10.1631/jzus.2006.A1077


Almost split sequences for symmetric non-semisimple Hopf algebras


Author(s):  SHI Mei-hua

Affiliation(s):  Department of Mathematics, Zhejiang Education Institute, Hangzhou 310012, China

Corresponding email(s):   smh@zjei.net

Key Words:  Indecomposable, Unimodular, Almost split sequences, Symmetric non-semisimple Hopf algebras


SHI Mei-hua. Almost split sequences for symmetric non-semisimple Hopf algebras[J]. Journal of Zhejiang University Science A, 2006, 7(6): 1077-1083.

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author="SHI Mei-hua",
journal="Journal of Zhejiang University Science A",
volume="7",
number="6",
pages="1077-1083",
year="2006",
publisher="Zhejiang University Press & Springer",
doi="10.1631/jzus.2006.A1077"
}

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T1 - Almost split sequences for symmetric non-semisimple Hopf algebras
A1 - SHI Mei-hua
J0 - Journal of Zhejiang University Science A
VL - 7
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Y1 - 2006
PB - Zhejiang University Press & Springer
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DOI - 10.1631/jzus.2006.A1077


Abstract: 
We first prove that for a finite dimensional non-semisimple Hopf algebra H, the trivial H-module is not projective and so the almost split sequence ended with k exists. By this exact sequence, for all indecomposable H-module X, we can construct a special kind of exact sequence ending with it. The main aim of this paper is to determine when this special exact sequence is an almost split one. For this aim, we restrict H to be unimodular and the square of its antipode to be an inner automorphism. As a special case, we give an application to the quantum double D(H)=(Hop)*H) of any non-semisimple Hopf algebra.

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Reference

[1] Auslander, M., Reiten, I., Smalø, S.O., 1995. Representation Theory of Artin Algebras. Cambridge University Press, Cambridge.

[2] Böhm, G., Nill, F., Szlachanyi, K., 1999. Weak Hopf algebras I. Integral theory and C*-structure. J. Algebra, 221(2):385-438.

[3] Kassel, C., 1995. Quantum Groups. GTM 155. Springer-Verlag, p.127-128.

[4] Lorenz, M., 1997. Representation of finite dimensinal Hopf algebras. J. Algebra, 188(2):476-505.

[5] Montgomery, S., 1993. Hopf Algebras and Their Actions on Rings. CBMS, Lecture in Math, Providence, RI, 82:215-217.

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