An Application of Finite Groups to Hopf algebras

Keywords: Hopf algebras, Integral elements, Semisimple Hopf algebra, Left coideal subalgebra, Solvability of groups and Hopf algebras, Nilpotency of groups and Hopf algebras.

Abstract

Kaplansky’s famous conjectures about generalizing results from groups to Hopf al-
gebras inspired many mathematicians to try to find solusions for them. Recently, Cohen and Westreich in [8] and [10] have generalized the concepts of nilpotency and solvability of groups to Hopf algebras under certain conditions and proved interesting results. In this article, we follow their work and give a detailed example by considering a finite group G and an algebraically closed field K. In more details, we construct the group Hopf algebra H = KG and examine its properties to see what of the properties of the original finite group can be carried out in the case of H.

Author Biographies

Tahani Al-Mutairi, Department of Mathematics, Qassim University, Buraydah, Saudi Arabia
Department of Mathematics, Qassim University, Buraydah, Saudi Arabia
Mohammed Mosa Al-shomrani, King Abdulaziz University
Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia;
Published
2021-08-05
Section
Nonlinear Analysis