黑克代數

設

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

黑克環${\displaystyle {\mathfrak {H}}_{l}(\epsilon )}$產生自：

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

而${\displaystyle \sigma _{i}}$要符合：

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

當l=1時，就約定${\displaystyle {\mathfrak {H}}_{1}(\epsilon )=\mathbb {C} }$。

留意：最後一項條件中當${\displaystyle \epsilon =1}$時，${\displaystyle \sigma _{i}^{2}=1}$，此所謂形變。

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