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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.

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  • $\begingroup$ That is because the map is not a homomorphism of algebras in general. To see this consider for example the algebra $k[x]/(x^2-2x)$ and the obvious basis (and see what happens when you square either side). $\endgroup$ Nov 3 '16 at 22:20
  • $\begingroup$ @TobiasKildetoft OP never said she wants $A$ to be a bialgebra, though. $\endgroup$
    – Pedro Tamaroff
    Nov 3 '16 at 22:24
  • $\begingroup$ @PedroTamaroff Ohh, good point (and of course the example I gave is actually a Hopf algebra with the given comultiplication as long as the basis is chosen differently). $\endgroup$ Nov 3 '16 at 22:25
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    $\begingroup$ Right, I was thinking of bialgebra. I have never seen coalgebras studied on their own (even though reps of algebraic groups are technically comodules for a coalgebra, the fact that it is really a Hopf algebra is needed in most places). $\endgroup$ Nov 3 '16 at 22:29
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    $\begingroup$ @Eli: the example is correct but no use is made of the algebra structure on $A$ so why mention it? $\endgroup$ Oct 3 '17 at 3:09
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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.

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