I have a problem with Sweedler notation, specifically in a example of the lecture notes:

There is an example/exercise that says:

The counity property is:

$\sum c_{(1)}\otimes\varepsilon\left(c_{(2)}\right)=\sum\varepsilon\left(c_{(1)}\right)\otimes c_{(2)} =c$

It shoud be:

$\sum c_{(1)}\varepsilon\left(c_{(2)}\right) =\sum\varepsilon\left(c_{(1)}\right)c_{(2)}=c$

Am I right, or they are equivalent?

1. There’s some abuse of notation here. It’s the kind of abuse of notation one gets used to, but it can be confusing at first.

(1) First of all, to get this sorted out: the counity property doesn’t just say $\sum c_{(1)}\epsilon\left(c_{(2)}\right)=\sum\epsilon\left(c_{(1)}\right)c_{(2)}$, but it says a bit more: it says $\sum c_{(1)}\epsilon\left(c_{(2)}\right)=\sum\epsilon\left(c_{(1)}\right)c_{(2)} = c$. I assume you know this, but I just wanted to make sure.

(2) Literally an equality like $\sum c_{(1)} \otimes \epsilon (c_{(2)}) = \sum\epsilon(c_{(1)}) \otimes c_{(2)}$ makes no sense, since its left hand side lives in $C\otimes \mathbb K$, while its right hand side lives in $\mathbb K \otimes C$.

(3) BUT there are some very convenient isomorphisms around between the vector spaces (and even algebras) $C$, $\mathbb K \otimes C$ and $C\otimes \mathbb K$. These isomorphisms allow us to identify these three vector spaces with each other. Under this identification, for instance, a tensor $\lambda \otimes c \in \mathbb K \otimes C$ becomes identified with the vector $\lambda c \in C$. So the equation $\sum c_{(1)}\epsilon\left(c_{(2)}\right)=\sum\epsilon\left(c_{(1)}\right)c_{(2)} = c$ can indeed be rewritten as $\sum c_{(1)}\otimes \epsilon\left(c_{(2)}\right)=\sum\epsilon\left(c_{(1)}\right)\otimes c_{(2)} = c$. Although I wouldn’t write it this way…

• Sebastián O.

thanks Darij for your answer! In (1) you’re right, I had this problem with the LaTeX code, so I accidentally skipped the “= c”. Sorry.

My doubt is exactly (3). Given those Isomorphism, it is “convinient” to write that (I think it’s wrong, for the same argument). And using it, could be an abuse of notation. Maybe it’s a problem based on the Sweedler notation.

• I hear you, Sebastian. But sometimes we sacrifice precision in order to write cleaner expressions.
For instance, in problem 2(b) in the homework, the left hand side lives in $C$ while the right hand side lives in $\mathbb{F} \otimes \mathbb{F} \otimes C$. We could get a more precise statement (or – fair enough – a true statement) if we wrote the left hand side as $1 \otimes 1 \otimes c$. But that’s uglier, so we don’t.
Microsoft would not call this “wrong”, it would call this a “feature” of the notation. 🙂 It is something to get used to.

2. Sebastián O.

So it’s correct to affirm, for instance, in exercise 2(b) that

$\sum c_{(1)}\otimes\varepsilon\left(c_{(2)}\right) = \sum c_{(1)}\varepsilon\left(c_{(2)}\right)$ ?

• Under the identification of $C \otimes {\mathbb{F}}$ with $C$, that is correct.