Misplaced Pages

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

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

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. arXiv:q-alg/9708022. Bibcode:1997CzJPh..47.1241M. doi:10.1023/A:1022877616496.

This linear algebra-related article is a stub. You can help Misplaced Pages by expanding it.

• v
• t
• e

• Vector spaces
• Lie algebras
• Linear algebra stubs