W -Algebras via Lax Type Operators
191
4 Quantum Finite W -Algebras and (Twisted) Yangians
4.1 Lax Type Operators for Quantum Finite W -Algebras
We introduce some important End V -valued polynomials in z, and Laurent series in
z −1 , with coefficients in U(g). The first one is (cf. (11))
A(z) = z1 V + U = z1 V +
i∈I
u i U
i
∈ U(g)[z] ⊗ End(V ) .
(17)
(As in Sect. 3, we are dropping the tensor product sign.) Another important operator
is (keeping the same notation as in [19])
A
ρ (z) = z1 V + F + π ≤
1
2
U = z1 V + F +
i∈I ≤ 1
2
u i U
i
∈ U(g)[z]⊗End V . (18)
Now we introduce the Lax operator L(z). Consider the generalized quasideterminant (cf. (16))
L(z) = |A
ρ (z) + D| V [
d
2 ],V [−
d
2 ] =
1 V [−
d
2 ]
z1 V + F + π ≤
1
2
U + D
−1 1 V [
d
2 ]
−1
,
(19)
where 1 V [−
d
2 ] and 1 V [
d
2 ] are defined in Sect. 3.2 (using the obvious splittings of
V given by the grading (9)), A ρ (z) is defined in Eq. (18) and D is the “shift
matrix” (12).
Let us denote by ¯
1 the image of 1 ∈ U(g) in the quotient U(g)
U(g){m −
(f |m)
m ∈ g ≥1 }. The Lax operator L(z) is defined as the image of
L(z) in this
quotient:
L(z) = L g,f,V (z) :=
L(z) ¯
1 .
(20)
The first main result in [21] can be summarized as follows.
Theorem 1
(a) The operator A ρ (z) + D is invertible in U(g)((z −1 )) ⊗ End V , and the
operator 1 V [−
d
2 ] (A ρ (z)+D) −1 1 V [
d
2 ] is invertible in U(g)((z −1 ))⊗Hom
V
−
d
2
, V
d
2
. Hence, the quasideterminant defining
L(z) (cf. (19)) exists and lies
in U(g)((z −1 )) ⊗ Hom
V
−
d
2
, V
d
2
.
(b) The entries of the coefficients of the operator L(z) defined in (20) lie in the
W -algebra W (g, f ):
L(z) : = |z1 V + F + π ≤
1
2
U + D| V [
d
2 ],V [−
d
2 ]
¯
1