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Crossed product of Hopf algebras

The main properties of the crossed product in the category of Hopf algebras are investigated. Let $A$ and $H$ be two Hopf algebras connected by two morphism of coalgebras $\triangleright : H\ot A \to A$, $f:H\ot H\to A$. The crossed product $A #_{f}^{\triangleright} H$ is a new Hopf algebra containing $A$ as a normal Hopf subalgebra. Furthermore, a Hopf algebra $E$ is isomorphic as a Hopf algebra to a crossed product of Hopf algebras $A #_{f}^{\triangleright} H$ if and only if $E$ factorizes through a normal Hopf subalgebra $A$ and a subcoalgebra $H$ such that $1_{E} \in H$. The universality of the construction, the existence of integrals, commutativity or involutivity of the crossed product are studied. Looking at the quantum side of the construction we shall give necessary and sufficient conditions for a crossed product to be a coquasitriangular Hopf algebra. In particular, all braided structures on the monoidal category of $A #_{f}^{\triangleright} H$-comodules are explicitly described in terms of their components. As an example, the braidings on a crossed product between $H_{4}$ and $k[C_{3}]$ are described in detail.