1 A multi-operator extension of the Jacobi identity In this section, we work over an arbitrary field of characteristic zero. (In Sections 2 and 3, it will be more convenient to work over C.) The symbols z ^o? zi, etc., shall designate independent commuting formal variables. 1.1 Vertex operator algebras In this subsection we recall a basic definition: A vertex operator algebra (cf. [6], Section 8.10) is a Z-graded vector space V = U y (n)5 n = wtv for v G V(n), (1.1) such that dimV(n) oo for n G Z, (1.2) V(n) = 0 for n sufficiently small, (1-3) equipped with a linear map V —*(EndV)[[z,*-1]] v*-*Y(v,z) = J^n*-"-1 (1.4) (vn G EndV), and with two distinguished homogeneous vectors l,u G V, satisfying the following conditions for u , u £ V : uni = 0 for n sufficiently large, (1-5) or, equivalently, Y(l,* ) = l (1.6) y(w,z) l V[[z]] and l i m r ^ 0 * , z ) l = v (1.7) ^ ( ^ ) y ( y ( u , ^ , : 2 ) (1.8) Z 2
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