The generalized smash product and coproduct (Q1586107)

It is shown in this article that some well known constructions of algebras made up from two algebras with certain additional structure can be understood in terms of generalized smash products. Explicitly, for two bialgebras \(A\) and \(H\) with an invertible skew pairing \(\tau\colon A\otimes H\to k\), a bialgebra \(A\bowtie_\tau H\) can be constructed by twisting the multiplication of \(A\otimes H\) by \(\tau\) (the Drinfeld double is an example of this construction). Another construction is \(A*H\), for \(H\) a Hopf algebra and \(A\) an \(H\)-bimodule algebra. On the other hand, let \(H\) be a bialgebra, \(A\) a left \(H\)-module algebra and \(X\) a left \(H\)-comodule algebra. The generalized smash product of \(A\) and \(X\) is the tensor product \(A\#X\) with the multiplication twisted by \(xa=\sum(x_{(-1)}a)x_{(0)}\). A first result explains when \(A\#X\) is a bialgebra and when it is a Hopf algebra. Then, it is proved that \(A\bowtie_\tau H\) as well as \(A*H\) can be constructed as generalized smash coproducts. A partial reciprocal statement establishes that if \(A\#X\) is a coquasitriangular bialgebra then it can be constructed as \(A\bowtie_\tau X\) for a given \(\tau\) (and \(A\) and \(X\) are coquasitriangular). At the end of the article, these results are dualized.