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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Doi-Hopf modules, Yetter-Drinfel'd modules
and Frobenius type properties


Authors: S. Caenepeel, G. Militaru and Shenglin Zhu
Journal: Trans. Amer. Math. Soc. 349 (1997), 4311-4342
MSC (1991): Primary 16W30
DOI: https://doi.org/10.1090/S0002-9947-97-02004-7
MathSciNet review: 1443189
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Abstract: We study the following question: when is the right adjoint of the forgetful functor from the category of $(H,A,C)$-Doi-Hopf modules to the category of $A$-modules also a left adjoint? We can give some necessary and sufficient conditions; one of the equivalent conditions is that $C\otimes A$ and the smash product $A\# C^*$ are isomorphic as $(A, A\# C^*)$-bimodules. The isomorphism can be described using a generalized type of integral. Our results may be applied to some specific cases. In particular, we study the case $A=H$, and this leads to the notion of $k$-Frobenius $H$-module coalgebra. In the special case of Yetter-Drinfel'd modules over a field, the right adjoint is also a left adjoint of the forgetful functor if and only if $H$ is finite dimensional and unimodular.


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Additional Information

S. Caenepeel
Affiliation: Faculty of Applied Sciences, University of Brussels, VUB, Pleinlaan 2, B-1050 Brussels, Belgium
Email: scaenepe@vnet3.vub.ac.be

G. Militaru
Affiliation: Faculty of Mathematics, University of Bucharest, Str. Academiei 14, RO-70109 Bucharest 1, Romania
Email: gmilitaru@roimar.imar.ro

Shenglin Zhu
Affiliation: Faculty of Mathematics, Fudan University, Shanghai 200433, China
Email: slzhu@ms.fudan.edu.cn

DOI: https://doi.org/10.1090/S0002-9947-97-02004-7
Keywords: Hopf algebras, Doi-Hopf modules, Yetter-Drinfel\textprime d modules, Frobenius extensions
Received by editor(s): May 9, 1995
Additional Notes: The second and the third author both thank the University of Brussels for its warm hospitality during their visit there.
Article copyright: © Copyright 1997 American Mathematical Society