Is every algebra a coalgebra?

Question

I was looking to study the topic of coalgebras and few examples were given. In particular, I didn't see anywhere this example: let $A$ be any $\mathbb{K}$-algebra, in particular it is a $\mathbb{K}$ vector space, let $\{a_i\}_{i \in I}$ be a base, then $A$ is a coalgebra with coproduct $\Delta(a_i) = a_i \otimes a_i$ and $\epsilon = 1$. I have seen this example but applied only to very specific algebras (like polynomials). It seems odd that is never said that any algebra is canonically a coalgebra.

Answer 1

Yes.
Your observation is correct and it is in fact valid in an even more general setting. It actually does not have to be a $k$-algebra, it works for any set:

Let $S$ be a non-empty set; $kS$ is the $k$ vector space with basis $S$. Then $kS$ is a coalgebra with comultiplication $\Delta$ and counit $\epsilon$ defined by $\Delta(s)=s\otimes s$, $\ \epsilon(s)=1$ for any $s\in S$.

This shows in particular, that any $k$-vector space can be endowed with a coalgebra structure (as has already been mentioned in a comment above by user Ender Wiggins).

The above form of the example is cited in various sources in the literature of coalgebras and Hopf algebras. See for example the book "Hopf algebras, an introduction" by Dascalescu, Nastasescu, Raianu. There, it is mentioned as an introductory example: See Example 1.1.4, 1)., p.3.