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Separable functors for the category of Doi-Hopf modules, applications. (English) Zbl 0943.18007

The authors give necessary and sufficient conditions for the functor that forgets the comodule structure from a category of Doi-Hopf modules to be a separable functor. This is used to obtain Maschke type theorems. The results are applied to Yetter-Drinfeld modules, Long dimodules and modules graded by \(G\)-sets.

MSC:

18E99 Categorical algebra
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
16D90 Module categories in associative algebras
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[1] Abe, E., Hopf Algebras (1977), Cambridge Univ. Press: Cambridge Univ. Press Cambridge
[2] Caenepeel, S.; Militaru, G.; Zhu, Shenglin, Crossed modules and Doi-Hopf modules, Israel J. Math., 100, 221-247 (1997) · Zbl 0888.16018
[3] Caenepeel, S.; Militaru, G.; Zhu, Schenglin, Doi-Hopf modules, Yetter-Drinfel’d modules and Frobenius type properties, Trans. Amer. Math. Soc., 349, 4311-4342 (1997) · Zbl 0890.16020
[4] Caenepeel, S.; Militaru, G.; Zhu, Shenglin, A Maschke type theorem for Doi-Hopf modules and applications, J. Algebra, 187, 388-412 (1997) · Zbl 0873.16024
[5] S. Caenepeel, Bogdan, Ion, G. Militaru, and, M. Stănciulescu, Notes on the separability equation, preprint, 1997.; S. Caenepeel, Bogdan, Ion, G. Militaru, and, M. Stănciulescu, Notes on the separability equation, preprint, 1997.
[6] Caenepeel, S.; Raianu, S., Induction functors for the Doi-Koppinen unified Hopf modules, Abelian Groups and Modules (1995), Kluwer Academic: Kluwer Academic Dordrecht, p. 73-94 · Zbl 0843.16035
[7] Cohen, M.; Fischman, D., Hopf algebra actions, J. Algebra, 100, 363-379 (1986) · Zbl 0591.16005
[8] Cohen, M.; Fischman, D., Semisimple extensions and elements of trace, 1, J. Algebra, 149, 419-437 (1992) · Zbl 0788.16029
[9] S. Dăscălescu, Bogdan, Ion, C. Năstăsescu, and, J. Rios Montes, Comodule algebra structures of matrix rings: Gradings, preprint, 1998.; S. Dăscălescu, Bogdan, Ion, C. Năstăsescu, and, J. Rios Montes, Comodule algebra structures of matrix rings: Gradings, preprint, 1998.
[10] DeMeyer, F.; Ingraham, E., Separable Algebras over Commutative Rings. Separable Algebras over Commutative Rings, Lecture Notes in Mathematics, 181 (1971), Springer-Verlag: Springer-Verlag Berlin · Zbl 0215.36602
[11] Doi, Y., Algebras with total integrals, Comm. Algebra, 13, 2137-2159 (1986) · Zbl 0576.16004
[12] Doi, Y., Hopf extensions of algebras and Maschke type theorems, Israel J. Math., 72, 99-108 (1990) · Zbl 0731.16025
[13] Doi, Y., Unifying Hopf modules, J. Algebra, 153, 373-385 (1992) · Zbl 0782.16025
[14] Doi, Y.; Takeuchi, M., Hopf-Galois extensions of algebras, the Miyashita-Ulbrich action and Azumaya algebras, J. Algebra, 121, 488-516 (1989) · Zbl 0675.16004
[15] Drinfel’d, V. G., Quantum groups, Proceedings. Proceedings, Internat. Congr. Math., 1 (1986), p. 789-820 · Zbl 0617.16004
[16] Hirata, K.; Sugano, K., On semisimple and separable extensions over noncommutative rings, J. Math. Soc. Japan, 18, 360-373 (1966) · Zbl 0178.36802
[17] Koppinen, M., Variations on the smash product with applications to group-graded rings, J. Pure Appl. Algebra, 104, 61-80 (1995) · Zbl 0838.16035
[18] Larson, R. G., Coseparable coalgebras, J. Pure Appl. Algebra, 3, 261-267 (1973) · Zbl 0276.18014
[19] Larson, R. G.; Radford, D. E., Finite dimensional cosemisimple Hopf algebras in characteristic 0 are semisimple, J. Algebra, 117, 267-289 (1988) · Zbl 0649.16005
[20] Long, F. W., The Brauer group of dimodule algebras, J. Algebra, 31, 559-601 (1974) · Zbl 0282.16007
[21] Larson, R. G.; Sweedler, M. E., An associative orthogonal bilinear form for Hopf algebras, Amer. J. Math., 91, 75-93 (1969) · Zbl 0179.05803
[22] Majid, S., Physics for algebrists: Non-commutative and non-cocommutative Hopf algebras by a bycrossproduct construction, J. Algebra, 130, 17-64 (1990) · Zbl 0694.16008
[23] Militaru, G., Functors for relative Hopf modules. Applications, Rev. Roumaine Math. Pures Appl., 41, 451-512 (1996) · Zbl 0928.16027
[24] Militaru, G., The Long dimodule category and nonlinear equations, Algebras and Representation Theory, 2, 1-24 (1999)
[25] Montgomery, S., Hopf Algebras and Their Actions on Rings (1993), Amer. Math. Soc: Amer. Math. Soc Providence · Zbl 0804.16041
[26] Năstăsescu, C.; Raianu, S.; van Oystaeyen, F., Modules graded by \(G\)-sets, Math. Z., 203, 605-627 (1990) · Zbl 0721.16025
[27] Năstăsescu, C.; van den Bergh, M.; van Oystaeyen, F., Separable functors applied to graded rings, J. Algebra, 123, 397-413 (1989) · Zbl 0673.16026
[28] Rafael, M. D., Separable functors revisited, Comm. Algebra, 18, 1445-1459 (1990) · Zbl 0713.18002
[29] del Rı́o, A., Categorical methods in graded ring theory, Public. Math., 72, 489-531 (1990) · Zbl 0781.16027
[30] Radford, D., Minimal quasi-triangular Hopf algebras, J. Algebra, 157, 285-315 (1993) · Zbl 0787.16028
[31] Radford, D., The order of the antipode in a finite-dimensional Hopf algebra is finite, Amer. J. Math., 98, 333-355 (1976) · Zbl 0332.16007
[32] Pareigis, B., When Hopf algebras are Frobenius algebras, J. Algebra, 18, 588-596 (1971) · Zbl 0225.16008
[33] Schneider, H.-J., Principal homogeneous spaces for arbitrary Hopf algebras, Israel J. Math., 72, 167-195 (1990) · Zbl 0731.16027
[34] D. Ştefan, and, F. Van Oystaeyen, The Wedderburn-Malcev theorem for comodule algebras, Comm. Algebra, to appear.; D. Ştefan, and, F. Van Oystaeyen, The Wedderburn-Malcev theorem for comodule algebras, Comm. Algebra, to appear. · Zbl 0945.16032
[35] Sweedler, M. E., Hopf Algebras (1969), Benjamin: Benjamin New York
[36] Takeuchi, M., Finite-dimensional representation of the quantum Lorenz group, Comm. Math. Phys., 144, 557-580 (1992) · Zbl 0747.17022
[37] Van Oystaeyen, F.; Xu, Y.; Zhang, Y., Induction and coinduction for Hopf extension, Sci. China Ser. B, 39, 246-263 (1996)
[38] Whitehead, J. H.C., Combinatorial homotopy, II, Bull. Amer. Math. Soc., 55, 453-496 (1949) · Zbl 0040.38801
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