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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.