# Anyonic Lie algebra

Short description: Graded vector space equipped with a bilinear operator

In mathematics, an anyonic Lie algebra is a U(1) graded vector space $\displaystyle{ L }$ over $\displaystyle{ \Complex }$ equipped with a bilinear operator $\displaystyle{ [\cdot, \cdot] \colon L \times L \rightarrow L }$ and linear maps $\displaystyle{ \varepsilon \colon L \to \Complex }$ (some authors use $\displaystyle{ |\cdot| \colon L \to \Complex }$) and $\displaystyle{ \Delta \colon L \to L\otimes L }$ such that $\displaystyle{ \Delta X = X_i \otimes X^i }$, satisfying following axioms:[1]

• $\displaystyle{ \varepsilon([X,Y]) = \varepsilon(X)\varepsilon(Y) }$
• $\displaystyle{ [X, Y]_i \otimes [X, Y]^i = [X_i, Y_j] \otimes [X^i, Y^j] e^{\frac{2\pi i}{n} \varepsilon(X^i) \varepsilon(Y_j)} }$
• $\displaystyle{ X_i \otimes [X^i, Y] = X^i \otimes [X_i, Y] e^{\frac{2 \pi i}{n} \varepsilon(X_i) (2\varepsilon(Y) + \varepsilon(X^i)) } }$
• $\displaystyle{ [X, [Y, Z]] = X i, Y], [X^i, Z e^{\frac{2 \pi i}{n} \varepsilon(Y) \varepsilon(X^i)} }$

for pure graded elements X, Y, and Z.

## References

1. Majid, S. (21 Aug 1997). "Anyonic Lie Algebras". Czechoslov. J. Phys. 47 (12): 1241–1250. doi:10.1023/A:1022877616496. Bibcode1997CzJPh..47.1241M.