黑克代数

设

• ${\displaystyle ϵ\in \mathbb{C}}${\displaystyle \epsilon \in \mathbb {C} }  
• ${\displaystyle l\geq 1}$ 

黑克环${\displaystyle {\mathfrak {H}}_{l}(\epsilon )}$ 产生自：

• ${\displaystyle \sigma _{1},\sigma _{2},......,\sigma _{l-1}}$ 

而${\displaystyle \sigma _{i}}$ 要符合：

• ${\displaystyle \sigma _{i}\sigma _{i}^{-1}=\sigma _{i}^{-1}\sigma _{i}=1}$ 
• 当|i-j|>1，就有 ${\displaystyle \sigma _{i}\sigma _{j}=\sigma _{j}\sigma _{i}}$ 
• 当j=i+1，就有${\displaystyle \sigma _{i}\sigma _{j}\sigma _{i}=\sigma _{j}\sigma _{i}\sigma _{j}}$ 
• ${\displaystyle (\sigma _{i}+1)(\sigma _{i}-\epsilon )=0}$ 

当l=1时，就约定${\displaystyle {\mathfrak {H}}_{1}(\epsilon )=\mathbb {C} }$ 。

留意：最后一项条件中当${\displaystyle \epsilon =1}$ 时，${\displaystyle \sigma _{i}^{2}=1}$ ，此所谓形变。

• Vyjayanthi Chari / Andrew Pressley (1994), "A Guide to Quantum Groups", Cambridge, ISBN 0-521-55884-0 , p.332