In mathematics, an anyonic Lie algebra is a U(1) graded vector space L {\displaystyle L} over C {\displaystyle \mathbb {C} } equipped with a bilinear operator [ ⋅ , ⋅ ] : L × L → L {\displaystyle [\cdot ,\cdot ]\colon L\times L\rightarrow L} and linear maps ε : L → C {\displaystyle \varepsilon \colon L\to \mathbb {C} } (some authors use | ⋅ | : L → C {\displaystyle |\cdot |\colon L\to \mathbb {C} } ) and Δ : L → L ⊗ L {\displaystyle \Delta \colon L\to L\otimes L} such that Δ X = X i ⊗ X i {\displaystyle \Delta X=X_{i}\otimes X^{i}} , satisfying following axioms:[1]
for pure graded elements X, Y, and Z.
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equipped with a bilinear operator ![{\displaystyle [\cdot ,\cdot ]\colon L\times L\rightarrow L}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70b0fe829f4fcaaf79f28b954195fe498f7172fa)
and linear maps 
(some authors use 
) and 
such that 
, satisfying following axioms:[1]
![{\displaystyle \varepsilon ([X,Y])=\varepsilon (X)\varepsilon (Y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33b25ef2e3af7983c3beed9d9b88fa0c14789d05)
![{\displaystyle [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})}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f228e7639ea84dd13b09a4a21e1d0152b2c917a5)
![{\displaystyle 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}))}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93a03a14d20cd62723ceab0f8349db7fc870a5e3)
![{\displaystyle [X,[Y,Z]]=[[X_{i},Y],[X^{i},Z]]e^{{\frac {2\pi i}{n}}\varepsilon (Y)\varepsilon (X^{i})}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db9bd3cb954e2ad56e2051410a6ae9ba4d0cb35c)
