Anyonic Lie algebra

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]

• ε ( [ X , Y ] ) = ε ( X ) ε ( Y ) {\displaystyle \varepsilon ([X,Y])=\varepsilon (X)\varepsilon (Y)}
• [ X , Y ] i ⊗ [ X , Y ] i = [ X i , Y j ] ⊗ [ X i , Y j ] e 2 π i n ε ( X i ) ε ( Y j ) {\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})}}
• X i ⊗ [ X i , Y ] = X i ⊗ [ X i , Y ] e 2 π i n ε ( X i ) ( 2 ε ( Y ) + ε ( X i ) ) {\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}))}}
• [ X , [ Y , Z ] ] = [ [ X i , Y ] , [ X i , Z ] ] e 2 π i n ε ( Y ) ε ( X i ) {\displaystyle [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.

References