TITLE: Exercises to help a student become accustomed to Sweedler notation
QUESTION [10 upvotes]: For a coassociative coalgebra $A$, we have a comultiplication map $\Delta \colon A \to A \otimes A$. An element $c \in A$ is sent to a sum of simple tensors, which can be a mess of indices, so we can use Sweedler's notation to try to write down the comultiplication more cleanly:
$$
  \Delta(c) = \sum_i c_{{(1)}_i} \otimes c_{{(2)}_i} 
  = \sum_{(c)} c_{{(1)}} \otimes c_{{(2)}}
$$
Sometimes even the sigma is omitted in the notation, and we simply write $\Delta(c) = c_{(1)} \otimes c_{(2)}$. Working with this sumless Sweedler notation takes some getting used to. What are some good introductory exercises to give a student practice working with Sweedler notation? I assume that there must be some canonical exercises like this one that only use the very basic properties of bialgebras or Hopf algebras.

REPLY [3 votes]: Prove that if a bialgebra $H$ admits a $k$-linear antipode map $S\colon H \to H$, then that map must be unique. 

This is a good exercise to work out in Sweedler notation, and uses only basic axioms of a Hopf algebra
$$ 
S(h_{(1)})h_{(2)} = \epsilon(h) = h_{(1)}S(h_{(2)}) 
\quad\text{and}\quad 
\epsilon(h_{(1)})h_{(2)} = h = h_{(1)}\epsilon(h_{(2)}) 
\,.
$$