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The structure of Hopf algebras with a projection. (English) Zbl 0549.16003
Let $$H$$ be a bialgebra over a field, and let $$B$$ be both a left $$H$$-module algebra and a left $$H$$-comodule coalgebra. Then $$B\otimes H$$ has the smash product structure of an algebra and the smash coproduct structure of a coalgebra (for a description of these structures, see R. Molnar [J. Algebra 47, 29-51 (1977; Zbl 0353.16004)]). The author investigates when this situation yields a bialgebra structure on $$B\otimes H$$. In this case, he calls $$(H,B)$$ an admissible pair. He shows that this situation is characterized by the canonical algebra injections of $$B$$ and $$H$$ into $$B\otimes H$$ and the canonical coalgebra projections of $$B\otimes H$$ onto $$B$$ and $$H$$. For $$(H,B)$$ an admissible pair, he studies when $$B\otimes H$$ is a Hopf algebra, the integrals of $$B\otimes H$$, and when $$B\otimes H$$ is semisimple or cosemisimple. He determines sufficient conditions for $$(H,B)$$ to be admissible when $$H$$ is a Hopf algebra in terms of the canonical mappings involving $$H$$ and $$B\otimes H$$. Finally, he studies the special case where $$H$$ is a group algebra and $$B$$ is a group-like coalgebra on a group, and uses this to construct some new examples of semisimple cosemisimple involutory Hopf algebras.
Reviewer: E.J.Taft

##### MSC:
 16W30 Hopf algebras (associative rings and algebras) (MSC2000)
##### MathOverflow Questions:
Integrals and finite dimensionality in braided Hopf algebras
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##### References:
 [1] Molnar, R.K, Semidirect products of Hopf algebras, J. algebra, 47, 29-51, (1977) · Zbl 0353.16004 [2] Radford, D.E, Finiteness conditions for a Hopf algebra with a non-zero integral, J. algebra, 46, 189-195, (1977) · Zbl 0361.16002 [3] Radford, D.E, Operators on Hopf algebras, Amer. J. math., 99, 1, 139-158, (1977) · Zbl 0369.16011 [4] Sullivan, J.B, The uniqueness of the integral for Hopf algebras and some existence theorems of integrals for commutative Hopf algebras, J. algebra, 19, 426-440, (1971) · Zbl 0239.16006 [5] Sweedler, M.E, Hopf algebras, (1969), Benjamin New York · Zbl 0194.32901
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