mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, an

anyon In physics, an anyon is a type of quasiparticle that occurs only in two-dimensional systems, with properties much less restricted than the two kinds of standard elementary particles, fermions and bosons. In general, the operation of exchangi ...

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...

is a ''U''(1)

graded vector space In mathematics, a graded vector space is a vector space that has the extra structure of a '' grading'' or a ''gradation'', which is a decomposition of the vector space into a direct sum of vector subspaces. Integer gradation Let \mathbb be th ...

L

over

\Complex

equipped with a

bilinear operator In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example. Definition Vector spaces Let V, W ...

\colon L \times L \rightarrow L

and

linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...

\varepsilon \colon L \to \Complex

(some authors use

, \cdot,  \colon L \to \Complex

) and

\Delta \colon L \to L\otimes L

such that

\Delta X = X_i \otimes X^i

, satisfying following axioms: *

= \varepsilon(X)\varepsilon(Y)

, Y The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...

i \otimes

i = _i, Y_j\otimes ^i, Y^je^ *

X_i \otimes^i, Y = X^i \otimes_i, Y e^

,_[Y,_Z_=_X_i,_Y.html"_;"title=",_Z.html"_;"title=",_[Y,_Z">,_[Y,_Z_=_X_i,_Y">,_Z.html"_;"title=",_[Y,_Z">,_[Y,_Z_=_X_i,_Y_[X^i,_Z.html" ;"title=",_Z">,_[Y,_Z_=_X_i,_Y.html" ;"title=",_Z.html" ;"title=", [Y, Z">, [Y, Z = X_i, Y">,_Z.html" ;"title=", [Y, Z">, [Y, Z = X_i, Y [X^i, Z">,_Z">,_[Y,_Z_=_X_i,_Y.html" ;"title=",_Z.html" ;"title=", [Y, Z">, [Y, Z = X_i, Y">,_Z.html" ;"title=", [Y, Z">, [Y, Z = X_i, Y [X^i, Z e^

for pure graded elements ''X'', ''Y'', and ''Z''.

References

Vector spaces Lie algebras {{algebra-stub