Hopf Algebra Structure Over Crossed Coproducts

The smash coproduct coalgebra has been generalized to crossed coproduct coalgebra in 3]. It is natural to replace the smash coproduct by the crossed coproduct and consider the conditions under which the smash product algebra structure and the crossed coproduct coalgebra structure will inherit a bialgebra structure or a Hopf algebra structure. We derive necessary and sufficient conditions for this problem. This generalizes the corresponding results in 7]. Finally, we characterize this new structure by introducing a concept of (H, )-comodule and prove that Heisenberg double 4] and smash coproduct do not make a bialgebra.AMS Subject Classification (1991): 16S40 16W30The first and second authors were partially supported by NNSF of China. The first author was also supported by the NSF of Henan Province and the second author by the YSF of Shandong Province (No. Q98A05113).