Anyonic Lie algebra

Short description: Graded vector space equipped with a bilinear operator

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

• $$\epsilon ([X,Y])=\epsilon (X)\epsilon (Y)$$
• $$[X,Y{]}_i\otimes [X,Y{]}^i=[X_i,Y_j]\otimes [X^i,Y^j]e^{\frac{2\pi i}{n}\epsilon (X^i)\epsilon (Y_j)}$$
• $$X_i\otimes [X^i,Y]=X^i\otimes [X_i,Y]e^{\frac{2\pi i}{n}\epsilon (X_i)(2\epsilon (Y)+\epsilon (X^i))}$$
• $$[X,[Y,Z]]=Xi,Y],[X^i,Z{e}^{\frac{2\pi i}{n}\epsilon (Y)\epsilon (X^i)}$$

for pure graded elements X, Y, and Z.