# Dual of an Algebra

I want to know that, as I know if I have two algebras and the morphism between them is an algebra morphism ... i am trying to figure out dual of that algebra morphism. For this I just need to take the dual of algebras and reverse the algebra map. Right! But I am having trouble in finding out the dual of my algebras. I know in general how to find the dual space of the vector space . But I couldn’t figure out Defoe some specific algebras . For example $$M_2(\mathbb{C})$$ two by two matrices of complex numbers also what will be the dual space for quaternion algebraquaternions. I will be thankful if you try to help me in this regard .

• It is not entirely clear what you mean here by the dual of an algebra. Is it just the dual as a vector space ? Is it also supposed to have an algebra structure ? – Captain Lama Apr 19 at 19:26
• It’s is just like the dual as vector space and having algebra structure also – Sania Asif Apr 19 at 19:28
• The dual of an algebra is normally a coalgebra. It does not naturally have an algebra structure unless you assume something extra. – Michal Adamaszek Apr 19 at 20:08
• What does your mean by assuming something extra? – Sania Asif Apr 19 at 20:11
• If $A$ is a $k$-algebra, you could define the dual of $A$ as a $k$-algebra as $\mathrm{Hom}_k(A,k)$, the algebra of $k$-algebra homomorphisms from $A$ to $k$ with pointwise operations. Is that what you mean? – Arturo Magidin Apr 19 at 21:36