184
D. Valeri
to a Hamiltonian functional
h ∈ V/∂V is the evolution equation
du
dt
= {
h, u} , u ∈ V .
(2)
The minimal requirement for integrability is to have an infinite collection of linearly
independent integrals of motion in involution:
h 0 =
h,
h 1 ,
h 2 , . . . s.t. {
h m ,
h n } = 0 for all m, n ∈ Z ≥0 .
In this case, we have the integrable hierarchy of Hamiltonian equations
du
dt n
= {
h n , u} , n ∈ Z ≥0 .
Example 1 The Virasoro–Magri PVA on the algebra of differential polynomials
V = C[u, u , u , . . . ] is defined by letting
{u λ u} = (2λ + ∂)u + λ
3 ,
and extending it to a λ-bracket for the whole V using sesquilinearity and Leibniz
rules. Let
h =
u 2
2 . Then the corresponding Hamiltonian equation (2) is the
famous KdV equation:
du
dt
= u
+ 3uu
.
Using the Lenard–Magri scheme of integrability [42] it can be shown that it belongs
to an integrable hierarchy.
Vertex Algebras
VAs were introduced in [6]. Following [14], we provide here a “Poisson-like”
definition using λ-brackets. A VA is a (not necessarily commutative nor associative)
unital algebra V with a derivation ∂ endowed with a λ-bracket [· λ ·] : V × V −→
V [λ] satisfying sesquilinearity, skewsymmetry, Jacobi identity, and, moreover
(a, b, c ∈ V ):
1. quasicommutativity: ab − ba =
0
−∂ [a λ b]dλ;
2. quasiassociativity: (ab)c − a(bc) = (| λ=∂ a)
λ
0 [b μ c]dμ + (| λ=∂ b)
λ
0 [a μ c]dμ;
3. noncommutative Wick formula: [a λ bc] = [a λ b]c + b[a λ c] +
λ
0 [[a λ b] μ c]dμ.
We refer to [14] for explanations about the notation. As before, we denote by
: V → V /∂V the canonical quotient map of vector spaces. If V is a VA, then
V /∂V carries a well-defined Lie algebra structure given by [
f,
g] =
[f λ g]| λ=0 ,
D. Valeri
to a Hamiltonian functional
h ∈ V/∂V is the evolution equation
du
dt
= {
h, u} , u ∈ V .
(2)
The minimal requirement for integrability is to have an infinite collection of linearly
independent integrals of motion in involution:
h 0 =
h,
h 1 ,
h 2 , . . . s.t. {
h m ,
h n } = 0 for all m, n ∈ Z ≥0 .
In this case, we have the integrable hierarchy of Hamiltonian equations
du
dt n
= {
h n , u} , n ∈ Z ≥0 .
Example 1 The Virasoro–Magri PVA on the algebra of differential polynomials
V = C[u, u , u , . . . ] is defined by letting
{u λ u} = (2λ + ∂)u + λ
3 ,
and extending it to a λ-bracket for the whole V using sesquilinearity and Leibniz
rules. Let
h =
u 2
2 . Then the corresponding Hamiltonian equation (2) is the
famous KdV equation:
du
dt
= u
+ 3uu
.
Using the Lenard–Magri scheme of integrability [42] it can be shown that it belongs
to an integrable hierarchy.
Vertex Algebras
VAs were introduced in [6]. Following [14], we provide here a “Poisson-like”
definition using λ-brackets. A VA is a (not necessarily commutative nor associative)
unital algebra V with a derivation ∂ endowed with a λ-bracket [· λ ·] : V × V −→
V [λ] satisfying sesquilinearity, skewsymmetry, Jacobi identity, and, moreover
(a, b, c ∈ V ):
1. quasicommutativity: ab − ba =
0
−∂ [a λ b]dλ;
2. quasiassociativity: (ab)c − a(bc) = (| λ=∂ a)
λ
0 [b μ c]dμ + (| λ=∂ b)
λ
0 [a μ c]dμ;
3. noncommutative Wick formula: [a λ bc] = [a λ b]c + b[a λ c] +
λ
0 [[a λ b] μ c]dμ.
We refer to [14] for explanations about the notation. As before, we denote by
: V → V /∂V the canonical quotient map of vector spaces. If V is a VA, then
V /∂V carries a well-defined Lie algebra structure given by [
f,
g] =
[f λ g]| λ=0 ,