Double crossed biproducts and related structures

TL;DR

This paper generalizes the construction of double crossed biproducts, providing new conditions for their algebraic and coalgebraic structures to form bialgebras, with applications to crossed product algebras.

Contribution

It improves the necessary conditions for double crossed biproducts to form bialgebras and introduces a more general two-sided crossed product algebra structure.

Findings

Established improved conditions for double crossed biproducts to be bialgebras.

Connected the condition to Majid's double biproduct framework.

Presented applications of the generalized crossed product 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 $\triangleright$. 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 $\triangleleft$. 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 $b_{[1]}\triangleright a_0\otimes b_{[0]}\triangleleft a_{-1}=a\otimes b$ 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 Brzez\'nski's crossed product and give some applications.