2.VERTEX OPERATOR ALGEBRAS
2.1. Formal calculus
We shall work in an algebraic setting over an arbitrary field F of characteristic
0. The set of nonnegative integers will be denoted by N. The symbols
Z,ZQ,Z\,
...
will designate commuting formal variables. For a vector space V", we set
V[z) = J2 vnZn\vn £ V, all but finitely many vn = 0 (2.1.1)
IneN J
V\z,z~x\ = ^2,vnzn\vn e V, all but finitely many vn - 0 (2.1.2)
Inez J
^[[*]] = | £ t ^ n e ^ } (2.1.3)
InGN J
V((z)) =
\Y1
v^zU\vn
eV, vn = 0 for sufficiently small n \ , (2.1.4)
Inez J
n M "
1
] ] - j ] » * » G v l , (2.1.5)
and we shall also use analogous notation for several variables.
The following formal version of Taylor's theorem is easily verified by direct
expansion: For f(z)
^vnzn
£ V[[z,
z'1)},
ez°-tf{z) = f{z + z0), (2.1.6)
where an expression of the form ex denotes the formal exponential series, and
where on the right-hand side, each binomial (z + ZQ)11 is understood to be ex-
panded in nonnegative integral powers of the second summand, namely, z$. We
shall repeatedly use this binomial expansion convention.
We introduce a basic generating function, the "formal (^-function at z 1" :
*0O = X
n
. (2.1.7)
nG2
9
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