Editing Vertex operator algebra (section)

==== Data ====
* a [[vector space]] <math>V</math>, called the space of states.  The underlying [[field (mathematics)|field]] is typically taken to be the [[complex number]]s, although Borcherds's original formulation allowed for an arbitrary [[commutative ring]].
* an identity element <math>1\in V</math>, sometimes written <math>|0\rangle</math> or <math>\Omega</math> to indicate a vacuum state.
* an [[endomorphism]] <math>T:V\rightarrow V</math>, called "translation". (Borcherds's original formulation included a system of divided powers of <math>T</math>, because he did not assume the ground ring was divisible.)
* a linear multiplication map <math>Y:V\otimes V\rightarrow V((z))</math>, where <math>V((z))</math> is the space of all [[formal Laurent series]] with coefficients in <math>V</math>. This structure has some alternative presentations:
** as an infinite collection of bilinear products <math> \cdot_n : u \otimes v \mapsto u_n v</math> where <math>n \in \mathbb{Z}</math> and <math> u_n \in \mathrm{End}(V)</math>, so that for each <math>v</math>, there is an <math>N</math> such that <math>u_n v = 0</math> for <math>n < N</math>.
** as a left-multiplication map <math>V\rightarrow \mathrm{End}(V)[[z^{\pm 1}]]</math>. This is the 'state-to-field' map of the so-called state-field correspondence. For each <math>u\in V</math>, the endomorphism-valued [[formal distribution]] <math>Y(u,z)</math> is called a vertex operator or a field, and the coefficient of <math>z^{-n-1}</math> is the operator <math>u_{n}</math>. In the context of vertex algebras, a '''field''' is more precisely an element of <math>\mathrm{End}(V)[[z^{\pm 1}]]</math>, which can be written <math>A(z) = \sum_{n \in \mathbb{Z}}A_n z^n, A_n \in \mathrm{End}(V)</math> such that for any <math>v \in V, A_n v = 0</math> for sufficiently small <math>n</math> (which may depend on <math>v</math>). The standard notation for the multiplication is
::<math display = block>u \otimes v \mapsto Y(u,z)v = \sum_{n \in \mathbf{Z}} u_n v z^{-n-1}.</math>