Maschke’s theorem for smash products of quasitriangular weak Hopf algebras

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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).

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).