Abstract
The paper is concerned with the semisimplicity of smash products of quasitriangular weak Hopf algebras. Let (H,R) be a finite dimensional quasitriangular weak Hopf algebra over a field k and A any semisimple and quantum commutative weak H-module algebra. Based on the work of Nikshych et al. (Topol. Appl. 127(1–2):91–123, 2003), we give Maschke’s theorem for smash products of quasitriangular weak Hopf algebras, stating that A#H is semisimple if and only if A is a projective left A#H-module, which extends the Theorem 3.2 given in Yang and Wang (Commun. Algebra 27(3):1165–1170, 1999).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alonso Álvarez, J.M., Fernández Vilaboa, J.N., González Rodríguez, R.: Weak Yang-Baxter operators and quasitriangular weak Hopf algebras. Arab. J. Sci. Eng. 33(2), 27–40 (2008)
Böhm, G., Nill, F., Szlachányi, K.: Weak Hopf algebras (I): Integral theory and C ∗-structure. J. Algebra 221(2), 385–438 (1999)
Caenepeel, S., Militaru, G., Zhu, S.: Frobenius and Separable Functors for Genneralized Hopf Modules and Nonlinear Equations. Lect. Notes in Math, vol. 1787. Springer, Berlin (2002)
Doi, Y.: On the structure of relative Hopf modules. Commun. Algebra 11(3), 243–255 (1983)
Etingof, P., Nikshych, D.: Dynamical quantum groups at roots of 1. Duke J. Math. 108, 135–168 (2001)
Etingof, P., Schiffmann, O.: Lectures on the dynamical Yang-Baxter equations. Lect. Notes Lond. Math. Soc. 290, 89–129 (2001)
Gao, N.: Actions of semisimple weak Hopf algebras. J. Math. Res. Expo. 28(1), 25–34 (2008)
Jia, L., Li, F.: Global dimension of weak smash product. J. Zhejiang Univ. Sci. A 7(12), 2088–2092 (2006)
Montgomery, S.: Hopf Algebras and their Actions on Rings. ICBMS, Chicago (1993)
Nikshych, D.: A duality theorem for quantum groupoids. Contemp. Math. 267, 237–243 (2000)
Nikshych, D., Vainerman, L.: Finite quantum groupoids and their applications. Math. Sci. Res. Inst. Publ. 43, 211–262 (2002)
Nikshych, D., Turaev, V., Vainerman, L.: Quantum groupoids and invariants of knots and 3-manifolds. Topol. Appl. 127(1–2), 91–123 (2003)
Wang, Y., Zhang, L.Y.: The structure theorem and duality theorem for endomorphism algebras of weak Hopf algebras. J. Pure Appl. Algebra (2010). doi:10.1016/j.jpaa.2010.07.015
Yang, S.L., Wang, Z.X.: The semisimplicity of smash products of quantum commutative algebras. Commun. Algebra 27(3), 1165–1170 (1999)
Zhang, L.Y.: Maschke-type theorem and Morita context over weak Hopf algebras. Sci. China Ser. A 49(5), 587–598 (2006)
Zhang, L.Y.: The structure theorem of weak Hopf algebras. Commun. Algebra 38(4), 1269–1281 (2010)
Zhang, L.Y., Zhu, S.L.: Fundamental theorems of weak Doi-Hopf modules and semisimple weak smash product Hopf algebras. Commun. Algebra 32(9), 3403–3415 (2004)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by I. Runkel.
Supported by the National Natural Science Foundation of China (10871170), the Educational Minister Science Technology Key Foundation of China (108154) and the College Special Research Doctoral Disciplines Point Fund of China (20100097110040).
Rights and permissions
About this article
Cite this article
Zhai, Wj., Zhang, Ly. Maschke’s theorem for smash products of quasitriangular weak Hopf algebras. Abh. Math. Semin. Univ. Hambg. 81, 35–44 (2011). https://doi.org/10.1007/s12188-010-0047-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12188-010-0047-7