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 𝒞. In the literature, there are equivalent conditions for (B,D) to form a generalized smash biproduct (or cross product) denoted by B× 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× B∗ when B has a left dual object B∗ in 𝒞. As results, we show that B× D is left smash over D× B∗, and establish a specific isomorphism B× DYD(𝒞)B× D≅D× B∗YD(𝒞)D× B∗ between the categories of the (left-right) Yetter-Drinfeld modules in 𝒞. The constructions and results are provided in three levels: (1) B× D is an algebra and a coalgebra; (2) B× D is a bialgebra; (3) B× D is a Hopf algebra.
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BibTeX
@misc{li2026characterization,
title = {{Characterization Results on Generalized Smash Biproduct Hopf Algebras over the Partial Dual Construction}},
author = {Kangqiao Li},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.39896},
eprint = {2609.39896},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39896},
}
RIS
TY - UNPB
TI - Characterization Results on Generalized Smash Biproduct Hopf Algebras over the Partial Dual Construction
AU - Kangqiao Li
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.39896
UR - https://arxiv.org/abs/2609.39896
AB - Let B and D be both algebras and coalgebras in a braided monoidal category 𝒞. In the literature, there are equivalent conditions for (B,D) to form a generalized smash biproduct (or cross product) denoted by B× 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× B^∗ when B has a left dual object B^∗ in 𝒞. As results, we show that B× D is left smash over D× B^∗, and establish a specific isomorphism _B× DYD(𝒞)^B× D≅_D× B^∗YD(𝒞)^D× B^∗ between the categories of the (left-right) Yetter-Drinfeld modules in 𝒞. The constructions and results are provided in three levels: (1) B× D is an algebra and a coalgebra; (2) B× D is a bialgebra; (3) B× D is a Hopf algebra.
ER -