194
D. Valeri
differential operators and pseudodifferential operators, respectively (see [20] for a
review of their basic properties).
Consider the differential operators
A(∂) = ∂1 V + U = ∂1 V +
i∈I
u i U
i
∈ V(g)[∂] ⊗ End(V )
and
A
ρ (∂) = ∂1 V + F + π ≤
1
2
U = ∂1 V + F +
i∈I ≤ 1
2
u i U
i
∈ V(g ≤
1
2
)[∂] ⊗ End V .
Recall from [15] that in the classical affine case we have W(g, f ) ⊂ V(g ≤
1
2
)
and that there exists a differential algebra isomorphism w : V(g f )
∼
−→ W(g, f ).
Consider the generalized quasideterminant (cf. (16))
L(∂) = |A
ρ (∂)| V [
d
2 ],V [−
d
2 ] =
1 V [−
d
2 ]
∂1 V + F + π ≤
1
2
U
−1 1 V [
d
2 ]
−1
.
(22)
The following result has been proved in [20].
Theorem 3 L(∂) ∈ W(g, f )((∂ −1 )) ⊗ Hom
V
−
d
2
, V
d
2
and
L(∂) =
1 V [−
d
2 ]
∂1 V + F +
i∈I f
w(u i )U
i
−1 1 V [
d
2 ]
−1
.
(23)
The above theorem consists of two statements. First, it claims that L(∂) is well
defined, i.e. both inverses in formula (22) can be carried out in the algebra of
pseudodifferential operators with coefficients in V(g ≤
1
2
), and that the coefficients
of L(∂) lie in the W-algebra W(g, f ). Then, it gives a formula, Eq. (23), for L(∂)
in terms of the generators w(u i ), i ∈ I f , of the W-algebra W(g, f ).
5.2 Integrable Hierarchies of Lax Type Equation
Let g be one of the classical Lie algebras gl N , sl N , so N or sp N , and let V = F N
be its standard representation (endowed, in the cases of so N and sp N , with a nondegenerate symmetric or skewsymmetric bilinear form, respectively). Then, we can
use the operator L(∂) in (23) to get explicit formulas for the λ-brackets among
the generators of W(g, f ) and construct integrable hierarchies of Hamiltonian
equations, see [20].