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 $[-,-]$ and linear maps $\varepsilon \colon L\to \mathbb {C}$ and $\Delta \colon L\to L\otimes L$ satisfying

$\varepsilon ([X,Y])=\varepsilon (X)\varepsilon (Y)$

for pure graded elements X, Y, and Z.