---
tag: 220PPH6N7Q
kind: definition
name: Sweedler notation
msc: [16T15, 16T05]
defines: NC26CE0Y8B
uses: [A06RA8KB0T, 7WEM2E6RTV, 0BF2DE2SYK]
sources:
  - url: https://en.wikipedia.org/wiki/Coalgebra
---

## Definition

Let $K$ be a field and let $(C, \Delta, \varepsilon)$ be a [[coalgebra|A06RA8KB0T]] over $K$.
For $c \in C$, the element $\Delta(c) \in C \otimes C$ can be written as a finite sum $\sum_{i=1}^{r} a_i \otimes b_i$ with $a_i, b_i \in C$.
*Sweedler notation* writes this as
$$\Delta(c) = \sum c_{(1)} \otimes c_{(2)},$$
and an expression $\sum f(c_{(1)}, c_{(2)})$ means $\sum_{i} f(a_i, b_i)$ for a map $f$ that is $K$-bilinear in its two arguments.
By coassociativity, the element $(\Delta \otimes \id)\Delta(c) = (\id \otimes \Delta)\Delta(c)$ is written $\sum c_{(1)} \otimes c_{(2)} \otimes c_{(3)}$.
For a right [[comodule|7WEM2E6RTV]] $M$ with coaction $\rho$, one writes $\rho(m) = \sum m_{(0)} \otimes m_{(1)}$, with $m_{(0)} \in M$ and $m_{(1)} \in C$.

## Remarks

Because $f$ is bilinear, $\sum_i f(a_i, b_i)$ depends only on $\Delta(c)$ and not on the chosen sum; it equals $\tilde f(\Delta(c))$ for the linear map $\tilde f$ on $C \otimes C$ induced by $f$.
Example: the counit law reads $\sum \varepsilon(c_{(1)}) c_{(2)} = c = \sum c_{(1)} \varepsilon(c_{(2)})$.
Example: for a group-like element $g$, with $\Delta(g) = g \otimes g$, one may take $g_{(1)} = g_{(2)} = g$.
Non-example: $\sum c_{(1)} \otimes c_{(1)}$ is not a valid expression, because it is not bilinear in $(c_{(1)}, c_{(2)})$.
For $\Delta(x) = x \otimes 1 + 1 \otimes x$ in $K[x]$ it would give $x \otimes x + 1 \otimes 1$.
The same element written as $(x + 1) \otimes 1 + 1 \otimes x - 1 \otimes 1$ would give $(x+1) \otimes (x+1) + 1 \otimes 1 + 1 \otimes 1$, a different result.