The New Dual of Smash Products, Braided Products and L-R Smash Products

Abstract:

      In this paper, we prove that the new dual $_H(A\# B)^0$ of the Smash product $A\# B$ introduced by dimodule algebras $A$ and $B$ is a Smash coproduct $_HA^0\times _HB^0$ introduced by dimodule coalgebras $_HA^0$ and $_HB^0$. If $(H,\sigma)$ is a braided Hopf algebra, we show that $_HH^0$ is a right, left $H^0$-dimodule coalgebra, and then prove that the new dual $_H(A\propto B)^0$ of the braided product $A\propto B$ introduced by quantum Yang-Baxter module algebras $A$ and $B$ is a braided coproduct $_HA^0\times _HB^0$ introduced by quantum Yang-Baxter module coalgebras $_HA^0$ and $_HB^0$. An exact sequence of the new dual ${(A\natural H)_H}^0$ of the L-R Smash product $A\natural H$ is given.