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 $R^{\underline {op} } \# H^{op}$ and $R^{\underline o } \# H^o$ respectively where $R^{\underline {op} }$ is a bialgebra in $_{H^{op} }^{H^{op} } \mathcal{Y}D$ and $R^{\underline o }$ 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.

Research by the first author partially supported by NSA Grant H98230-04-1-0061.