Editing Vertex operator algebra (section)

===Vertex operator algebra===
A '''vertex operator algebra''' is a vertex algebra equipped with a '''conformal element''' <math>\omega \in V</math>, such that the vertex operator <math>Y(\omega,z)</math> is the weight two Virasoro field <math>L(z)</math>:

:<math>Y(\omega, z) = \sum_{n\in\mathbf{Z}} \omega_{n} {z^{-n-1}} = L(z) = \sum_{n\in\mathbf{Z}} L_n z^{-n-2}</math>

and satisfies the following properties:
* <math>[L_m,L_n]=(m-n)L_{m+n}+\frac{1}{12}\delta_{m+n,0}(m^3-m)c\,\mathrm{Id}_V</math>, where <math>c</math> is a constant called the '''central charge''', or '''rank''' of <math>V</math>.  In particular, the coefficients of this vertex operator endow <math>V</math> with an action of the Virasoro algebra with central charge <math>c</math>.
* <math>L_0</math> acts semisimply on <math>V</math> with integer eigenvalues that are bounded below.
* Under the grading provided by the eigenvalues of <math>L_0</math>, the multiplication on <math>V</math> is homogeneous in the sense that if <math>u</math> and <math>v</math> are homogeneous, then <math>u_n v</math> is homogeneous of degree <math>\mathrm{deg}(u)+\mathrm{deg}(v)-n-1</math>.
* The identity <math>1</math> has degree 0, and the conformal element <math>\omega</math> has degree 2.
* <math>L_{-1}=T</math>.

A homomorphism of vertex algebras is a map of the underlying vector spaces that respects the additional identity, translation, and multiplication structure. Homomorphisms of vertex operator algebras have "weak" and "strong" forms, depending on whether they respect conformal vectors.