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

• $$\backslash varepsilon([X,Y])\; =\; \backslash varepsilon(X)\backslash varepsilon(Y)$$
• $$[X,\; Y]\_i\; \backslash otimes\; [X,\; Y]^i\; =\; [X\_i,\; Y\_j]\; \backslash otimes\; [X^i,\; Y^j]\; e^\{\backslash frac\{2\backslash pi\; i\}\{n\}\; \backslash varepsilon(X^i)\; \backslash varepsilon(Y\_j)\}$$
• $$X\_i\; \backslash otimes\; [X^i,\; Y]\; =\; X^i\; \backslash otimes\; [X\_i,\; Y]\; e^\{\backslash frac\{2\; \backslash pi\; i\}\{n\}$$

\varepsilon(X_i) (2\varepsilon(Y) + \varepsilon(X^i)) }

• $$[X,\; [Y,\; Z]]\; =X\_i,\; Y],\; [X^i,\; Ze^\{\backslash frac\{2\; \backslash pi\; i\}\{n\}\; \backslash varepsilon(Y)\; \backslash varepsilon(X^i)\}$$

for pure graded elements X, Y, and Z.