Anyonic Lie algebra

Last updated August 13, 2023 • 1 min readFrom Wikipedia, The Free Encyclopedia

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

$\varepsilon ([X,Y])=\varepsilon (X)\varepsilon (Y)$
$[X,Y]_{i}\otimes [X,Y]^{i}=[X_{i},Y_{j}]\otimes [X^{i},Y^{j}]e^{{\frac {2\pi i}{n}}\varepsilon (X^{i})\varepsilon (Y_{j})}$
$X_{i}\otimes [X^{i},Y]=X^{i}\otimes [X_{i},Y]e^{{\frac {2\pi i}{n}}\varepsilon (X_{i})(2\varepsilon (Y)+\varepsilon (X^{i}))}$
$[X,[Y,Z]]=[[X_{i},Y],[X^{i},Z]]e^{{\frac {2\pi i}{n}}\varepsilon (Y)\varepsilon (X^{i})}$

for pure graded elements X, Y, and Z.

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References

↑ Majid, S. (21 Aug 1997). "Anyonic Lie Algebras". Czechoslov. J. Phys. 47 (12): 1241–1250. arXiv: q-alg/9708022 . Bibcode:1997CzJPh..47.1241M. doi:10.1023/A:1022877616496.

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[1] Majid, S. (21 Aug 1997). "Anyonic Lie Algebras". Czechoslov. J. Phys. 47 (12): 1241–1250. arXiv: q-alg/9708022 . Bibcode:1997CzJPh..47.1241M. doi:10.1023/A:1022877616496.

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