# Matrices over quaternions make Hopf Algebra or not?

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.

• Please someone answer. I’m in great trouble ☹️ – Sania Asif Mar 14 at 6:03
• What is the bialgebra structure? Can you provide some context? – Joppy Mar 14 at 10:20
• Bialgebra means any vector space over field satisfying algebra and coalgebra axioms. I mean having unit counit multiplication and comultiplication. – Sania Asif Mar 14 at 10:28
• Yes, what is the specific comultiplication you have in mind? Even for matrices over the complex field? – Joppy Mar 14 at 10:31
• Counit maps the matrix to kronicker delta.,unit take quaternion and result into matrix representation of quaternion, comultiplication simply take a matrix in quaternion and split it into tw matrices . Antipode structure doesn’t satisfy if I take antipode to be adjoint of matrix.. I’m not sure about my thinking and I don’t know exactly how to write them in mathematical notation explicitly. – Sania Asif Mar 14 at 10:48