As we work with coalgebras, we’ll need a nice way to write out
the comultiplication of an element. In the group algebra we’ve
been using as an example, we just have , but not all elements
are so cleanly sent to two copies of themselves. And other
comltiplications in other coalgebras aren’t even defined so
nicely on any basis. So we introduce the so-called “Sweedler
notation”. If you didn’t like the summation convention, you’re
going to hate this. Okay, first of all, we know that the
comultiplication of an element is an element of the tensor
square . Thus it can be written as a finite sum Now, this uses
two whole new letters, and , which might be really awkward to
come up with in practice. Instead, let’s call them and , to
denote the first and second factors of the comultiplication.
We’ll also move the indices to superscripts, just to get them
out of the way. The whole index-summing thing is a bit awkward,
especially because the number of summands is different for each
coalgebra element . Let’s just say we’re