# 餘代數

## 定義

1. ${\displaystyle (\mathrm {id} _{C}\otimes \Delta )\circ \Delta =(\Delta \otimes \mathrm {id} _{C})\circ \Delta }$
2. ${\displaystyle (\mathrm {id} _{C}\otimes \epsilon )\circ \Delta =\mathrm {id} _{C}=(\epsilon \otimes \mathrm {id} _{C})\circ \Delta }$ .

## Sweedler 記法

${\displaystyle \Delta (c)=\sum _{i}c_{(1)}^{(i)}\otimes c_{(2)}^{(i)}.}$

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

${\displaystyle c=\sum _{(c)}\epsilon (c_{(1)})c_{(2)}=\sum _{(c)}c_{(1)}\epsilon (c_{(2)}).\;}$

${\displaystyle \sum _{(c)}c_{(1)}\otimes \left(\sum _{(c_{(2)})}(c_{(2)})_{(1)}\otimes (c_{(2)})_{(2)}\right)=\sum _{(c)}\left(\sum _{(c_{(1)})}(c_{(1)})_{(1)}\otimes (c_{(1)})_{(2)}\right)\otimes c_{(2)}.}$

${\displaystyle \sum _{(c)}c_{(1)}\otimes c_{(2)}\otimes c_{(3)}.}$

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

${\displaystyle c=\epsilon (c_{(1)})c_{(2)}=c_{(1)}\epsilon (c_{(2)}).\;}$

## 相關文獻

• Eiichi Abe, Hopf Algebras (1980), translated by Hisae Kinoshita and Hiroko Tanaka, Cambridge University Press. ISBN 0-521-22240-0