Characterization Results on Generalized Smash Biproduct Hopf Algebras over the Partial Dual Construction

Abstract

Let $B$ and $D$ be both algebras and coalgebras in a braided monoidal category $\mathcal{C}$. In the literature, there are equivalent conditions for $(B,D)$ to form a generalized smash biproduct (or cross product) denoted by $B\times D$, which were mainly given by Bespalov and Drabant in 1999 as well as by Bulacu, Caenepeel and Torrecillas in 2013. This paper is devoted to constructing another biproduct $D\times B^\ast$ when $B$ has a left dual object $B^\ast$ in $\mathcal{C}$. As results, we show that $B\times D$ is left smash over $D\times B^\ast$, and establish a specific isomorphism ${}_{B\times D}\mathfrak{YD}(\mathcal{C})^{B\times D}\cong{}_{D\times B^\ast}\mathfrak{YD}(\mathcal{C})^{D\times B^\ast}$ between the categories of the (left-right) Yetter-Drinfeld modules in $\mathcal{C}$. The constructions and results are provided in three levels: (1) $B\times D$ is an algebra and a coalgebra; (2) $B\times D$ is a bialgebra; (3) $B\times D$ is a Hopf algebra.