# 霍普夫代數

## 定義

${\displaystyle \forall c\in C,\quad S(c_{(1)})c_{(2)}=c_{(1)}S(c_{(2)})=\epsilon (c)1}$

## 例子

• ${\displaystyle \Delta :K[G]\to K[G]\otimes K[G],\quad \forall g\in G,\Delta (g)=g\otimes g}$
• ${\displaystyle \epsilon :K[G]\to K,\quad \forall g\in G,\epsilon (g)=1}$
• ${\displaystyle S:K[G]\to K[G],\quad \forall g\in G,S(g)=g^{-1}}$

• ${\displaystyle \Delta :K^{G}\to K^{G\times G},\quad \Delta (f)(x,y)=f(xy)}$
• ${\displaystyle \epsilon :K^{G}\to G,\quad \epsilon (f)=f(e)}$
• ${\displaystyle S:K^{G}\to K^{G},\quad S(f)(x)=f(x^{-1})}$

• ${\displaystyle \Delta :U\to U\otimes U,\quad \forall g\in {\mathfrak {g}},\Delta (x)=x\otimes 1+1\otimes x}$
• ${\displaystyle S:U\to U,\quad \forall x\in {\mathfrak {g}},S(x)=-x}$

## 李群的上同調

${\displaystyle H^{\bullet }(G)\to H^{\bullet }(G\times G)=H^{\bullet }(G)\otimes H^{\bullet }(G)}$

${\displaystyle A}$${\displaystyle K}$ 上的有限維分次交換、餘交換之霍普夫代數，則 ${\displaystyle A}$（視為 ${\displaystyle K}$-代數）同構於由奇數次元素生成的自由外代數

## 註記

1. ^ H. Hopf, Uber die Topologie der Gruppen-Mannigfaltigkeiten und ihrer Verallgemeinerungen, Ann. of Math. 42 (1941), 22-52. Reprinted in Selecta Heinz Hopf, pp. 119-151, Springer, Berlin (1964). MR4784