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Maschke’s theorem for smash products of quasitriangular weak Hopf algebras. (English) Zbl 1228.16034

Summary: 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 D. Nikshych, V. Turaev and L. Vainerman [Topology Appl. 127, No. 1-2, 91-123 (2003; Zbl 1021.16026)], 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 by S. Yang and Z. Wang [Commun. Algebra 27, No. 3, 1165-1170 (1999; Zbl 0927.16024)].

MSC:

16T05 Hopf algebras and their applications
16S40 Smash products of general Hopf actions
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[1] 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) · Zbl 1186.16019
[2] Böhm, G., Nill, F., Szlachányi, K.: Weak Hopf algebras (I): Integral theory and C structure. J. Algebra 221(2), 385–438 (1999) · Zbl 0949.16037 · doi:10.1006/jabr.1999.7984
[3] 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) · Zbl 1008.16036
[4] Doi, Y.: On the structure of relative Hopf modules. Commun. Algebra 11(3), 243–255 (1983) · Zbl 0502.16009 · doi:10.1080/00927878308822847
[5] Etingof, P., Nikshych, D.: Dynamical quantum groups at roots of 1. Duke J. Math. 108, 135–168 (2001) · Zbl 1023.17007 · doi:10.1215/S0012-7094-01-10814-4
[6] Etingof, P., Schiffmann, O.: Lectures on the dynamical Yang-Baxter equations. Lect. Notes Lond. Math. Soc. 290, 89–129 (2001) · Zbl 1036.17013
[7] Gao, N.: Actions of semisimple weak Hopf algebras. J. Math. Res. Expo. 28(1), 25–34 (2008) · Zbl 1153.16036
[8] Jia, L., Li, F.: Global dimension of weak smash product. J. Zhejiang Univ. Sci. A 7(12), 2088–2092 (2006) · Zbl 1121.16033 · doi:10.1631/jzus.2006.A2088
[9] Montgomery, S.: Hopf Algebras and their Actions on Rings. ICBMS, Chicago (1993) · Zbl 0793.16029
[10] Nikshych, D.: A duality theorem for quantum groupoids. Contemp. Math. 267, 237–243 (2000) · Zbl 0978.16032 · doi:10.1090/conm/267/04273
[11] Nikshych, D., Vainerman, L.: Finite quantum groupoids and their applications. Math. Sci. Res. Inst. Publ. 43, 211–262 (2002) · Zbl 1026.17017
[12] Nikshych, D., Turaev, V., Vainerman, L.: Quantum groupoids and invariants of knots and 3-manifolds. Topol. Appl. 127(1–2), 91–123 (2003) · Zbl 1021.16026 · doi:10.1016/S0166-8641(02)00055-X
[13] 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 · Zbl 1214.16028
[14] Yang, S.L., Wang, Z.X.: The semisimplicity of smash products of quantum commutative algebras. Commun. Algebra 27(3), 1165–1170 (1999) · Zbl 0927.16024 · doi:10.1080/00927879908826487
[15] Zhang, L.Y.: Maschke-type theorem and Morita context over weak Hopf algebras. Sci. China Ser. A 49(5), 587–598 (2006) · Zbl 1121.16036 · doi:10.1007/s11425-006-0587-6
[16] Zhang, L.Y.: The structure theorem of weak Hopf algebras. Commun. Algebra 38(4), 1269–1281 (2010) · Zbl 1197.16038 · doi:10.1080/00927870902849591
[17] 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) · Zbl 1073.13006 · doi:10.1081/AGB-120029915
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