User:Hillgentleman/Drinfeld twist

設

• ${\mathfrak {g}}$ be a simple Lie algebra
• $U{\mathfrak {g}}$ be its universal enveloping algebra
• $(H,m,\Delta ,i,\epsilon ,S)$ be the corresponding quantum universal enveloping algebra

定義 一下（Drinfeld所謂嘅）扭係元 $J\in H\otimes H$，滿足

• $J=1$(mod h)
• $(\epsilon \otimes 1)J=(1\otimes \epsilon )J=1$
• $J_{12,3}(J_{12}\otimes 1)(1\otimes J_{23})^{-1}J_{1,23}^{-1}\in (H^{\otimes 3})^{H}$ - $H\otimes H\otimes H$入面嘅 H-唔變量

我地有

命題： 每一扭都會由舊嘅Hopf 代數結構定義出一新嘅Hopf結構：

• ${\tilde {\Delta }}:=J^{-1}\Delta J$ - the new comultiplication
• ${\tilde {S}}:=Q^{-1}SQ$ - new antipode
• ${\tilde {\epsilon }}:=\epsilon$ - counit

參攷

• Etingof/Schiffmann, "lectures on quantum groups", p. 84