Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
monoidal Hom-Hopf algebra; separable functors; Maschke type theorem; total integral; relative Hom-Hopf module
monoidal Hom-Hopf algebra; separable functors; Maschke type theorem; total integral; relative Hom-Hopf module
Summary:
Let $(H,\alpha )$ be a monoidal Hom-Hopf algebra and $(A,\beta )$ a right $(H,\alpha )$-Hom-comodule algebra. We first introduce the notion of a relative Hom-Hopf module and prove that the functor $F $ from the category of relative Hom-Hopf modules to the category of right $(A, \beta )$-Hom-modules has a right adjoint. Furthermore, we prove a Maschke type theorem for the category of relative Hom-Hopf modules. In fact, we give necessary and sufficient conditions for the functor that forgets the $(H, \alpha )$-coaction to be separable. This leads to a generalized notion of integrals.
References:
[1] Caenepeel, S., Goyvaerts, I.: Monoidal Hom-Hopf algebras. Commun. Algebra 39 (2011), 2216-2240. DOI 10.1080/00927872.2010.490800 | MR 2813174 | Zbl 1255.16032
[2] Caenepeel, S., Militaru, G., Ion, B., Zhu, S.: Separable functors for the category of Doi-Hopf modules, applications. Adv. Math. 145 (1999), 239-290. DOI 10.1006/aima.1998.1817 | MR 1704577 | Zbl 0943.18007
[3] Doi, Y.: Hopf extensions of algebras and Maschke type theorems. Isr. J. Math. 72 (1990), 99-108. DOI 10.1007/BF02764613 | MR 1098982 | Zbl 0731.16025
[4] Doi, Y.: Algebras with total integrals. Commun. Algebra 13 (1985), 2137-2159. MR 0801433 | Zbl 0576.16004
[5] Doi, Y.: On the structure of relative Hopf modules. Commun. Algebra 11 (1983), 243-255. DOI 10.1080/00927878308822847 | MR 0688207 | Zbl 0502.16009
[6] Frégier, Y., Gohr, A.: On Hom-type algebras. J. Gen. Lie Theory Appl. 4 (2010), Article ID G101001, pages 16. DOI 10.4303/jglta/G101001 | MR 2795570 | Zbl 1281.17002
[7] Hartwig, J. T., Larsson, D., Silvestrov, S. D.: Deformations of Lie algebras using $\sigma$-derivations. J. Algebra 295 (2006), 314-361. DOI 10.1016/j.jalgebra.2005.07.036 | MR 2194957 | Zbl 1138.17012
[8] Makhlouf, A., Silvestrov, S.: Hom-algebras and Hom-coalgebras. J. Algebra Appl. 9 (2010), 553-589. DOI 10.1142/S0219498810004117 | MR 2718646 | Zbl 1259.16041
[9] Makhlouf, A., Silvestrov, S. D.: Hom-algebra structures. J. Gen. Lie Theory Appl. 2 (2008), 51-64. DOI 10.4303/jglta/S070206 | MR 2399415 | Zbl 1184.17002
[10] Takeuchi, M.: Relative Hopf modules-equivalences and freeness criteria. J. Algebra 60 (1979), 452-471. DOI 10.1016/0021-8693(79)90093-0 | MR 0549940 | Zbl 0492.16013
Search
Partner of
