Double crossed biproducts and related structures

Abstract

Let $H$ be a bialgebra. Let $\sigma :H\otimes H\to A$ be a linear map, where $A$ is a left $H$-comodule coalgebra, and an algebra with a left $H$-weak action $▹$. Let $\tau :H\otimes H\to B$ be a linear map, where $B$ is a right $H$-comodule coalgebra, and an algebra with a right $H$-weak action $◃$. In this paper, we improve the necessary conditions for the two-sided crossed product algebra $A{\#}^{\sigma }H{}^{\tau }\#B$ and the two-sided smash coproduct coalgebra $A\times H\times B$ to form a bialgebra (called double crossed biproduct) such that the condition $b_{[1]}▹a_0\otimes b_{[0]}◃a_{-1}=a\otimes b$ in Majid's double biproduct (or double-bosonization) is one of the necessary conditions. On the other hand, we provide a more general two-sided crossed product algebra structure via Brzezński's crossed product and give some applications.

Communications in Algebra,2022