Transactions of the American Mathematical Society

Author(s): S. Caenepeel; G. Militaru; Shenglin Zhu
Journal: Trans. Amer. Math. Soc. 349 (1997), 4311-4342.
MSC (1991): Primary 16W30
Retrieve article in: PDF
This article is available free of charge

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.

1.: F. W. Anderson, K. R. Fuller, ``Rings and categories of modules'', Springer-Verlag, New York, 1974.MR 54:5281
2.: S. Caenepeel, G. Militaru, S. Zhu, Crossed modules and Doi-Hopf modules, Israel J. Math., to appear.
3.: S. Caenepeel, \c{S}. Raianu, Induction functors for the Doi-Koppinen unified Hopf modules, in Abelian groups and Modules, Kluwer Academic Publishers, Dordrecht, 1995, pp. 73-94. MR 97a:16071
4.: S. D[??]asc[??]alescu, C. N[??]ast[??]asescu, A. Del Rio, F. van Oystaeyen, Gradings of finite support. Applications to injective objects, J. Pure and Appl. Algebra 107 (1996), 193-206. MR 97d:16046
5.: Y. Doi, Homological coalgebra, J. Math. Soc. Japan 33 (1981), 31-50.MR 82g:16014
6.: Y. Doi, Unifying Hopf modules, J. Algebra 153 (1992), 373-385.MR 94c:16048
7.: V. G. Drinfel'd, Quantum groups, Proc. Int. Cong. Math., Berkeley, 1 (1986), 789-820.MR 89f:17017
8.: C. Faith, Algebra: Rings, Modules and Categories I, Springer-Verlag, 1973. MR 51:3206
9.: R. Farnsteiner, On Frobenius extensions defined by Hopf algebras, J. Algebra 166 (1994), 130-141.MR 95e:16034
10.: D. Fischman, S. Montgomery, H.J. Schneider, Frobenius extensions of subalgebras of Hopf algebras, Trans. Amer. Math. Soc., to appear.
11.: M.A. Knus, M. Ojanguren, ``Théorie de la descente et algèbres d'Azumaya'', Lecture Notes in Math. 389, Springer Verlag, Berlin, 1974.MR 54:5209
12.: S. Majid, Physics for algebraists: non-commutative and non-cocommutative Hopf algebras by a bycrossproduct construction, J. Algebra 130 (1990), 17-64.MR 91j:16050
13.: C. Menini, C. N[??]ast[??]asescu, When are induction and coinduction functors isomorphic?, Bull. of the Belgian Math. Soc. Simon Stevin 1 (1994), 521-558.MR 96a:16041
14.: G. Militaru, From graded rings to actions and coactions of Hopf algebras, An. \c{S}tiin\c{t}. Univ. Ovidius Univ. of Constanta Ser. Mat. 2 (1994), 106-111.MR 97a:16074
15.: S. Montgomery, Hopf algebras and their actions on rings, CBMS Regional Conf. Ser. Math., no. 82, American Mathematical Society, Providence, 1993.MR 94i:16019
16.: C. N[??]ast[??]asescu, \c{S}. Raianu, F. van Oystaeyen, Modules graded by -sets, Math. Z. 203 (1990), 605-627.MR 91h:16071
17.: B. Pareigis, When Hopf algebras are Frobenius algebras, J. Algebra 18 (1971), 588-596.MR 43:6242
18.: D. Radford, Minimal quasi-triangular Hopf algebras, J. Algebra 157 (1993), 285-315.MR 94c:16052
19.: D. Radford, J. Towber, Yetter-Drinfel'd categories associated to an arbitrary bialgebra, J. Pure and Appl. Algebra 87 (1993), 259-279.MR 94f:16060
20.: M. E. Sweedler, Hopf algebras, Benjamin, New York, 1969. MR 40:5705
21.: M. Takeuchi, A correspondence between Hopf ideals and and sub-Hopf algebras, Manuscripta Math. 7 (1972), 251-270. MR 48:328
22.: K. H. Ulbrich, Smash products and comodules of linear maps, Tsukuba J. Math. 2 (1990), 371-378.MR 91m:16030
23.: D.N. Yetter, Quantum groups and representations of monoidal categories, Math. Proc. Camb. Phil. Soc. 108 (1990), 261-290.MR 91k:16028
24.: B. Zhou, Hopf-Doi data, functors: applications I, II and III, Comm. Algebra, to appear.

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 16W30

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: 10.1090/S0002-9947-97-02004-7
PII: S 0002-9947(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.
Copyright of article: Copyright 1997, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS