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)

Keywords:

• bimodule algebras
• L-R smash products
• coquasitriangular Hopf algebras
• Long bialgebras

Notes

* 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)