Английская Википедия:Anyonic Lie algebra

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

• <math display="block">\varepsilon([X,Y]) = \varepsilon(X)\varepsilon(Y)</math>
• <math display="block">[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 display="block">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 display="block">[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.