On Smash Products Of Hopf Algebras

Abstract

Let H = X R A denote an R-smash product of the two bialgebras X and A. We prove that X A is a pair of matched bialgebras, if the R-smash product H has a braiding structure. When X is an associative algebra and A is a Hopf algebra, we investigate the global dimension and the weak dimension of the smash product H = X R A and show that lD H ≤ rD A + lD X and wD H ≤ wD A + wD X . As an application, we get lD H 4 = for Sweedler's four dimensional Hopf algebra H 4 . We also study the associativity of smash products and the relations between smash products and factorization for algebras.