Anyonic Lie algebra

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

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

for pure graded elements X, Y, and Z.