L-R smash products for bimodule algebras*

Abstract In this paper, we prove that the L-R smash product A ♮ H is exactly the twisted smash product A * H if H is a finite dimensional cocommutative Hopf algebra, and give a sufficient and necessary condition for L-R smash products to be bialgebras (Hopf algebras). For any finite dimensional coquasitriangular Hopf algebra (H, σ), we prove that the L-R smash product H ♮ H is semisimple Artinian if H is semisimple and H* is unimodular. In particular, the L-R smash product D(H)* ♮D(H) * semisimple Artinian if the Drinfel’d double D(H) is semisimple. * Supported by National Natural Science Foundation of China (Grant No. 10571153), the Postdoctoral Science Foundation of China (Grant No. 2005037713), and the Postdoctoral Science Foundation of Jiangsu (Grant No. 0203003403)