Anyonic Lie algebra explained

L over \Complex equipped with a bilinear operator [ ⋅ , ⋅ ] \colon L x L → L and linear maps \varepsilon \colon L \to \Complex (some authors use | ⋅ | \colon L \to \Complex ) and \Delta \colon L \to L ⊗ L such that \Delta X = X _{i} ⊗ X ^{i} , satisfying following axioms:^[1]

• $$\backslash varepsilon([X,Y])\; =\; \backslash varepsilon(X)\backslash varepsilon(Y)$$
• $$[X,\; Y]\_i\; \backslash otimes\; [X,\; Y]^i\; =\; [X\_i,\; Y\_j]\; \backslash otimes\; [X^i,\; Y^j]\; e^$$
• $$X\_i\; \backslash otimes\; [X^i,\; Y]\; =\; X^i\; \backslash otimes\; [X\_i,\; Y]\; e^$$
• $$[X,\; [Y,\; Z]]\; =\; X\_i,\; Y,\; [X^i,\; Z]]\; e^$$

for pure graded elements X, Y, and Z.

References

1. Majid. S.. 21 Aug 1997. Anyonic Lie Algebras. Czechoslov. J. Phys.. 47. 12. 1241–1250. q-alg/9708022. 10.1023/A:1022877616496. 1997CzJPh..47.1241M.