I am learning Hopf algebra now a days. I am still confused about it’s axioms. I don’t know how to define antipode structure. What are the basic rules to define it. ? Can any one help me to solve this problem that matrices over quaternions are Hopf algebra or not?your answers will be highly appreciated. Even someone can also provide me the difference in the hopf algebra structure for Matrices over complex field and matrices over quaternions.
2026-03-25 15:51:04.1774453864
Matrices over quaternions make Hopf Algebra or not?
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Quaternions form an algebra over the reals, so I assume you look for real bialgebra structures. In particular you need a counit that must be an algebra map $M_n(\mathbb H)\to \mathbb R$. If you restrict to real matrices you get an algebra map from matrices to the reals. There are no such, unless n=1. Also, for n=1, there is no algebra map from quaternions to reals. So, there is no bialgebra structures on matrices over the quaternions.