# nLab Sweedler notation

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## 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 ${c}_{\left(1\right)}$ and ${c}_{\left(2\right)}$. Therefore

$\Delta \left(c\right)=\sum _{i=1}^{n}{c}_{\left(1\right)i}\otimes {c}_{\left(2\right)i}.$\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

$\Delta \left(c\right)=\sum {c}_{\left(1\right)}\otimes {c}_{\left(2\right)}$\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 gets more useful, when we take into account coassociativity to justify extending the notation to write

$\begin{array}{c}\sum {c}_{\left(1\right)}\otimes {c}_{\left(2\right)}\otimes {c}_{\left(3\right)}:=\sum {c}_{\left(1\right)\left(1\right)}\otimes {c}_{\left(1\right)\left(2\right)}\otimes {c}_{\left(3\right)}\\ =\sum {c}_{\left(1\right)}\otimes {c}_{\left(2\right)\left(1\right)}\otimes {c}_{\left(2\right)\left(2\right)}.\end{array}$\array{\sum c_{(1)}\otimes c_{(2)} \otimes c_{(3)} := \sum c_{(1)(1)}\otimes c_{(1)(2)}\otimes c_{(3)} \\ = \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 $\rho \left(v\right)=\sum {v}_{\left(0\right)}\otimes {v}_{\left(1\right)}$. Then we can use the coaction axiom $\left({\mathrm{id}}_{V}\otimes \Delta \right)\circ \rho =\left(\rho \otimes {\mathrm{id}}_{C}\right)\circ \rho$ to write

$\begin{array}{c}{v}_{\left(0\right)}\otimes {v}_{\left(1\right)}\otimes {v}_{\left(2\right)}:={v}_{\left(0\right)\left(0\right)}\otimes {v}_{\left(0\right)\left(1\right)}\otimes {v}_{\left(1\right)}\\ ={v}_{\left(0\right)}\otimes {v}_{\left(1\right)\left(0\right)}\otimes {v}_{\left(1\right)\left(1\right)},\end{array}$\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.

On big use is that the scalars like $ϵ\left({a}_{\left(3\right)}\right)$ 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.

Revised on February 26, 2013 23:11:29 by Gerrit Goosen? (105.236.28.114)