Anyonic Lie algebra

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

\varepsilon(X_i) (2\varepsilon(Y) + \varepsilon(X^i)) }$$
 * $$\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}
 * $$[X, [Y, Z]] = X_i, Y], [X^i, Z e^{\frac{2 \pi i}{n} \varepsilon(Y) \varepsilon(X^i)}$$

for pure graded elements X, Y, and Z.