Anyonic Lie algebra information

In mathematics, an anyonic Lie algebra is a U(1) graded vector space ${\displaystyle L}$ over ${\displaystyle \mathbb {C} }$ equipped with a bilinear operator ${\displaystyle [\cdot ,\cdot ]\colon L\times L\rightarrow L}$ and linear maps ${\displaystyle \varepsilon \colon L\to \mathbb {C} }$ (some authors use ${\displaystyle |\cdot |\colon L\to \mathbb {C} }$) and ${\displaystyle \Delta \colon L\to L\otimes L}$ such that ${\displaystyle \Delta X=X_{i}\otimes X^{i}}$, satisfying following axioms:^[1]

• $${\displaystyle \varepsilon ([X,Y])=\varepsilon (X)\varepsilon (Y)}$$

  
• $${\displaystyle [X,Y]_{i}\otimes [X,Y]^{i}=[X_{i},Y_{j}]\otimes [X^{i},Y^{j}]e^{{\frac {2\pi i}{n}}\varepsilon (X^{i})\varepsilon (Y_{j})}}$$

  
• $${\displaystyle X_{i}\otimes [X^{i},Y]=X^{i}\otimes [X_{i},Y]e^{{\frac {2\pi i}{n}}\varepsilon (X_{i})(2\varepsilon (Y)+\varepsilon (X^{i}))}}$$

  
• $${\displaystyle [X,[Y,Z]]=[[X_{i},Y],[X^{i},Z]]e^{{\frac {2\pi i}{n}}\varepsilon (Y)\varepsilon (X^{i})}}$$

  

for pure graded elements X, Y, and Z.