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Journal of Zhejiang University SCIENCE A 2006 Vol.7 No.12 P.2088-2092

http://doi.org/10.1631/jzus.2006.A2088


Global dimension of weak smash product


Author(s):  JIA Ling, LI Fang

Affiliation(s):  Department of Mathematics, Zhejiang University, Hangzhou 310027, China

Corresponding email(s):   jialing471@126.com

Key Words:  Weak Hopf algebra, Weak smash product, Gobal dimension


JIA Ling, LI Fang. Global dimension of weak smash product[J]. Journal of Zhejiang University Science A, 2006, 7(12): 2088-2092.

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author="JIA Ling, LI Fang",
journal="Journal of Zhejiang University Science A",
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number="12",
pages="2088-2092",
year="2006",
publisher="Zhejiang University Press & Springer",
doi="10.1631/jzus.2006.A2088"
}

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%DOI 10.1631/jzus.2006.A2088

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T1 - Global dimension of weak smash product
A1 - JIA Ling
A1 - LI Fang
J0 - Journal of Zhejiang University Science A
VL - 7
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SP - 2088
EP - 2092
%@ 1673-565X
Y1 - 2006
PB - Zhejiang University Press & Springer
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DOI - 10.1631/jzus.2006.A2088


Abstract: 
In Artin algebra representation theory there is an important result which states that when the order of G is invertible in Λ then gl.dim(ΛG)=gl.dim(Λ). With the development of Hopf algebra theory, this result is generalized to smash product algebra. As known, weak Hopf algebra is an important generalization of Hopf algebra. In this paper we give the more general result, that is the relation of homological dimension between an algebra A and weak smash product algebra A#H, where H is a finite dimensional weak Hopf algebra over a field k and A is an H-module algebra.

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Reference

[1] Auslander, M., 1995. Representation Theory of Artin Algebras. Cambridge Press.

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

[3] Böhm, G., 2000. Doi-Hopf modules over weak Hopf algebras. Comm. Algebra, 28:4687-4689.

[4] Nikshych, D., Vainerman, L., 2002. Finite Quantum Groupoids and Their Applications. In: Montgomery, S., Schneider, H.J. (Eds.), New Direction in Hopf Algebras, Series: Mathematical Sciences Research Institute Publications, Cambridge Univ. Press, Cambridge, 43:211-262.

[5] Nikshych, D., Turaev, V., Vainerman, L., 2003. Invariants of knots and 3-manifolds from quantum groupoids. Topology and its Applications, 127(1-2):91-123.

[6] Yang, S.L., 2002. Global dimension for Hopf actions. Comm. Algebra, 30(8):3653-3667.

[7] Zhang, L.Y., Zhu, S.L., 2004. Fundamental theorems of weak doi-Hopf modules and semisimple weak smash product Hopf algebras. Comm. Algebra, 32(9):3403-3416.

[8] Zhu, J.G., Zhang, L.Y., 2003. Semisimple extensions and homological dimension of invariants for Hopf actions. Acta Mathematica Sinica, English Series, 46(1):137-142.

[9] Zhu, J.G., 2005. Cosemisimple coextesnsion and homological dimension of smash coproduct. Acta Mathematica Sinica, English Series, 21(3):563-568.

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