L-R-smash biproducts, double biproducts and a braided category of Yetter-Drinfeld-Long bimodules

TL;DR

This paper introduces the concept of L-R-smash biproducts, generalizing Radford biproducts, and constructs a prebraided monoidal category of Yetter-Drinfeld-Long bimodules with applications in bialgebra theory.

Contribution

It defines L-R-smash biproducts, establishes conditions for their structure, and constructs a new prebraided monoidal category of bimodules with bialgebra properties.

Findings

L-R-smash biproducts generalize Radford biproducts.

Conditions for D to form a bialgebra in the category are identified.

A new prebraided monoidal category of bimodules is constructed.

Abstract

Let H be a bialgebra and D an H-bimodule algebra H-bicomodule coalgebra. We find sufficient conditions on D for the L-R-smash product algebra and coalgebra structures on D\otimes H to form a bialgebra (in this case we say that (H, D) is an L-R-admissible pair), called L-R-smash biproduct. The Radford biproduct is a particular case, and so is, up to isomorphism, a double biproduct with trivial pairing. We construct a prebraided monoidal category {\cal LR}(H), whose objects are H-bimodules H-bicomodules M endowed with left-left and right-right Yetter-Drinfeld module as well as left-right and right-left Long module structures over H, with the property that, if (H, D) is an L-R-admissible pair, then D is a bialgebra in {\cal LR}(H).