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 [math]\displaystyle{ L }[/math] over [math]\displaystyle{ \Complex }[/math] equipped with a bilinear operator [math]\displaystyle{ [\cdot, \cdot] \colon L \times L \rightarrow L }[/math] and linear maps [math]\displaystyle{ \varepsilon \colon L \to \Complex }[/math] (some authors use [math]\displaystyle{ |\cdot| \colon L \to \Complex }[/math]) and [math]\displaystyle{ \Delta \colon L \to L\otimes L }[/math] such that [math]\displaystyle{ \Delta X = X_i \otimes X^i }[/math], satisfying following axioms:

• [math]\displaystyle{ \varepsilon([X,Y]) = \varepsilon(X)\varepsilon(Y) }[/math]
• [math]\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)} }[/math]
• [math]\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)) } }[/math]
• [math]\displaystyle{ [X, [Y, Z]] = X i, Y], [X^i, Z e^{\frac{2 \pi i}{n} \varepsilon(Y) \varepsilon(X^i)} }[/math]

for pure graded elements X, Y, and Z.