nLab Sweedler notation

Idea

Sweedler notation is a special notation for discussion of operations in coalgebras

Definition

If $C$ is a coassociative coalgebra and then for $c\in C$, the comultiplication $\Delta$ maps $c$ to an element in $C\otimes C$ which is therefore a sum of the form $\sum_{i=1}^n a_i\otimes b_i$. Sweedler suggests that we do not make up new symbols like $a$ and $b$ but rather use composed symbols $c_{(1)}$ and $c_{(2)}$. Therefore

$$\Delta(c) = \sum_{i=1}^n c_{(1)i}\otimes c_{(2)i}.$$

Sweedler notation means that for certain manipulations involving just generic linear operations we actually do not need to think of the summation symbol $i$, so we can just write

$$\Delta(c) = \sum c_{(1)}\otimes c_{(2)}$$

with or even without summation sign. Surely in either case we need to remember that we do not have a factorization but we do have a sum of possibly more than one entry. One can formalize in fact which manipulations are allowed with such a reduced notation.

It becomes more useful when we take into account coassociativity to justify extending the notation to write

$$\array{\sum c_{(1)}\otimes c_{(2)} \otimes c_{(3)} := \sum c_{(1)(1)}\otimes c_{(1)(2)}\otimes c_{(2)} \\ = \sum c_{(1)}\otimes c_{(2)(1)}\otimes c_{(2)(2)}.}$$

Furthermore, we can extend it to coactions, e.g. $\rho:V\to V\otimes C$, by $\rho(v) = \sum v_{(0)}\otimes v_{(1)}$. Then we can use the coaction axiom $(id_V\otimes \Delta)\circ\rho = (\rho\otimes id_C)\circ \rho$ to write

$$\array{v_{(0)}\otimes v_{(1)} \otimes v_{(2)} := v_{(0)(0)}\otimes v_{(0)(1)} \otimes v_{(1)}\\ = v_{(0)}\otimes v_{(1)(0)}\otimes v_{(1)(1)},}$$

where we used the sumless Sweedler notation.

One big use is that the scalars like $\epsilon(a_{(3)})$ can be moved freely along the expression, which is difficult to write without calculating with Sweedler components: one would need lots of brackets and flip operators, and this could be messy and abstract.

References

The notation is named after Moss Sweedler. Sometimes (though rarely) it is also called Heyneman-Sweedler notation.