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On Smash Products Of Hopf Algebras
Authors:Ji Qingzhong  Qin Hourong
Institution:1. Department of Mathematics , Nanjing University , Nanjing, Jiangsu, China qingzhji@ouc.edu.cn;3. Department of Mathematics , Nanjing University , Nanjing, Jiangsu, China
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.
Keywords:Hopf algebra  Projective dimension  Smash products  Spectral sequence
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