Quasigroups, Braided Hopf (Co)quasigroups and Radford’s Biproducts of Quasi-Diagonal Type

Abstract

Given the Yetter–Drinfeld category over any quasigroup and a braided Hopf coquasigroup in this category, we first mainly study the Radford’s biproduct corresponding to this braided Hopf coquasigroup. Then, we investigate Sweedler’s duality of this braided Hopf coquasigroup and show that this duality is also a braided Hopf quasigroup in the Yetter–Drinfeld category, generalizing the main result in a Hopf algebra case of Ng and Taft’s paper. Finally, as an application of our results, we show that the space of binary linearly recursive sequences is closed under the quantum convolution product of binary linearly recursive sequences.

Keywords

• groups
• Yetter–Drinfel’s category
• braided Hopf (co)quasigroups
• symmetric category
• binary linearly recursive sequence