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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Doi-Hopf modules, Yetter-Drinfel’d modules and Frobenius type properties
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by S. Caenepeel, G. Militaru and Shenglin Zhu PDF
Trans. Amer. Math. Soc. 349 (1997), 4311-4342 Request permission


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:
  • G. Militaru
  • Affiliation: Faculty of Mathematics, University of Bucharest, Str. Academiei 14, RO-70109 Bucharest 1, Romania
  • Email:
  • Shenglin Zhu
  • Affiliation: Faculty of Mathematics, Fudan University, Shanghai 200433, China
  • Email:
  • 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.
  • © Copyright 1997 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 349 (1997), 4311-4342
  • MSC (1991): Primary 16W30
  • DOI:
  • MathSciNet review: 1443189