Anyonic Lie algebra: Difference between revisions
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In [[mathematics]], an '''[[anyon]]ic [[Lie algebra]]''' is a ''U''(1) [[graded vector space]] <math>L</math> over <math>\mathbb{C}</math> equipped with a [[bilinear operator]] <math>[\cdot, \cdot] \colon L \times L \rightarrow L</math> and [[linear map]]s <math>\varepsilon\colon L\to\mathbb{C}</math> (some authors use <math>|\cdot|\colon L\to \mathbb{C}</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:<ref>{{Cite journal|last=Majid|first=S.|date=21 Aug 1997|title=Anyonic Lie Algebras|journal=Czechoslov. J. Phys.|volume=47|issue=12|pages=1241–1250|arxiv=q-alg/9708022|doi=10.1023/A:1022877616496|bibcode=1997CzJPh..47.1241M}}</ref> |
In [[mathematics]], an '''[[anyon]]ic [[Lie algebra]]''' is a ''U''(1) [[graded vector space]] <math>L</math> over <math>\mathbb{C}</math> equipped with a [[bilinear operator]] <math>[\cdot, \cdot] \colon L \times L \rightarrow L</math> and [[linear map]]s <math>\varepsilon\colon L\to\mathbb{C}</math> (some authors use <math>|\cdot|\colon L\to \mathbb{C}</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:<ref>{{Cite journal|last=Majid|first=S.|date=21 Aug 1997|title=Anyonic Lie Algebras|journal=Czechoslov. J. Phys.|volume=47|issue=12|pages=1241–1250|arxiv=q-alg/9708022|doi=10.1023/A:1022877616496|bibcode=1997CzJPh..47.1241M}}</ref> |
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Revision as of 02:33, 13 May 2020
In mathematics, an anyonic Lie algebra is a U(1) graded vector space over equipped with a bilinear operator and linear maps (some authors use ) and such that , satisfying following axioms:[1]
![{\displaystyle [\cdot ,\cdot ]\colon L\times L\rightarrow L}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70b0fe829f4fcaaf79f28b954195fe498f7172fa)
![{\displaystyle \varepsilon ([X,Y])=\varepsilon (X)\varepsilon (Y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33b25ef2e3af7983c3beed9d9b88fa0c14789d05)
![{\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})}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f228e7639ea84dd13b09a4a21e1d0152b2c917a5)
![{\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}))}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93a03a14d20cd62723ceab0f8349db7fc870a5e3)
![{\displaystyle [X,[Y,Z]]=[[X_{i},Y],[X^{i},Z]]e^{{\frac {2\pi i}{n}}\varepsilon (Y)\varepsilon (X^{i})}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db9bd3cb954e2ad56e2051410a6ae9ba4d0cb35c)
for pure graded elements X, Y, and Z.
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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