Biproducts and Two-Cocycle Twists of Hopf Algebras

Abstract

Let $H$ be a Hopf algebra with bijective antipode over a field $k$ and suppose that $R{#}H$ is a bi-product. Then $R$ is a bialgebra in the Yetter--Drinfel'd category ${}_H^H\mathcal{Y}D$. We describe the bialgebras $(R{#}H)^{op}$ and $(R{#}H)^o$ explicitly as bi-products $R^{\UOP}{#}H^{op}$ and $R^{\UO}{#}H^o$ respectively where $R^{\UOP}$ is a bialgebra in ${}_{H^{op}}^{H^{op}}\mathcal{Y}D$ and $R^{\UO}$ is a bialgebra in ${}_{H^o}^{H^o}\mathcal{Y}D$. We use our results to describe two-cocycle twist bialgebra structures on the tensor product of bi-products.

amstex, 29 pages