W-ALGEBRAS AT THE CRITICAL LEVEL 5
The W -algebra associated with (g,f) at level k is by definition
Wk(g,f)
= Hf2

+0
(V
k(g)),
which is naturally a vertex algebra because d is the zero mode of a odd field d(z)
of C(V
k(g)).
We have [DSK] that, for any k,
RWk(g,f)

=
C[S], (11)
and (1) gives the isomorphism
C[S∞] gr
Wk(g,f),
(12)
see [A6].
Let k = cri. For z z(g), we have dz = 0, and the class of z belongs to the
center Z(Wcri(g,f)) of Wk(g,f). Hence the embedding z(g) V cri(g) induces the
vertex algebra homomorphism z(g) Z(Wcri(g,f)) Wcri(g,f).
Proposition 2.1. The embedding z(g) V cri(g) induces the embedding z(g)
Wcri(g,f).
Proof. It is sufficient to show that it induces an injective homomorphism
gr z(g) gr
Wcri(g,f).
Under the identification (6) and (11), the induced map
RV
cri(g)
RWcri(g,f) is identified the restriction map
C[g∗]G
C[S], and hence
is injective [Kos, P1]. Therefore it induces the injective map (RV
cri(g)
)∞
(RWcri(g,f))∞, which is identical to the map gr z(g) gr
Wcri(g,f).
By Proposition 2.1 we can define the quotient
cri(g,f)
of
Wcri(g,f)
for χ
OpLg
reg
as in Introduction. Let
Wres(g,f) =
Wχ0(g,f)cri
and call it the restricted W -algebra associated with (g,f). It is a graded quotient
of
Wcri(g,f).
Remark 2.2. Let Zhu(Wres(g,f)) be the Ramond twisted Zhu algebra [DSK]
of Wres(g,f). Then from Proposition 2.3 below it follows that
Zhu(Wres(g,f))

=
U(g,f)/Z(g)∗U(g,f),
where
Z(g)∗
is the argumentation ideal of Z(g).
Set S = S N as in Introduction. By restricting (4) and (5), we obtain the
isomorphisms
M × S

(f +
m⊥)
N , (13)
M∞ × S∞

((f +
m⊥)
N )∞ = (f +
m⊥)∞
N∞. (14)
Proposition 2.3. We have the following.
(i) H

2
+i
f
(Vres(g)) = 0 for i = 0 and H

2
+0
f
(Vres(g))

=
Wres(g,f) as vertex
algebras.
(ii)
Wcri(g,f)
is free over z(g).
(iii) RWres(g,f)

= C[S] as Poisson algebras and gr Wres(g,f)

= C[S∞] as ver-
tex Poisson algebras.
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