# Vertex algebras and algebraic curves by Frenkel E., Ben-Zwi D.

By Frenkel E., Ben-Zwi D.

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In contrast, the structure of g–modules Vk (g) for different values of k may be very different. 4), while V0 (g) is not. 4. Vertex algebra structure. We now wish to define a vertex algebra structure on the vacuum representation. ,dim g be an ordered basis of g. Split the extension g as a vector space. For any A ∈ g and n ∈ Z, we denote def An = A ⊗ tn ∈ Lg. Then the elements Jna , n ∈ Z, and K form a (topological) basis for g, while the elements Jna , n ≥ 0, and K form a basis for the “positive” subalgebra from which we induced Vk (g).

Lepowsky and A. Meurman [FLM] (see also [FHL, DL]). The works [B1] and [FLM] were motivated by the study of the Monster group. For an introduction to the subject, see the book [Kac3] by V. Kac. 24 1. DEFINITION OF VERTEX ALGEBRAS We adopt the system of axioms from [FKRW] and [Kac3]. The equivalence between these axioms and the original axioms of [B1, FLM] is explained in [Kac3] (see also [DL, Li1]). 3) first appeared in [DL], where it was shown that it can be used as a replacement for the original axioms of [B1, FLM] (see also [Li1]).

If A(z) and B(z) are homogeneous of conformal dimensions ∆B and ∆B , respectively, then :A(z)B(z): is also homogeneous of conformal dimension ∆A + ∆B . 32 2. VERTEX ALGEBRAS ASSOCIATED TO LIE ALGEBRAS (3) :A(w)B(w): = Resz=0 (δ(z − w)− A(z)B(w) + δ(z − w)+ B(w)A(z)). 4. Remark. In general, the operation of normally ordered product is neither commutative nor associative. 5. General fields. With normal ordering in hand we may now proceed to define our field Y (b2−1 , z) as Y (b2−1 , z) = :b(z)2 : .