W -Algebras via Lax Type Operators
185
and we have a representation of the Lie algebra V /∂V on V given by [
f, g] =
[f λ g]| λ=0 . A quantum integrable system consists in a collection of infinitely many
linearly independent elements
h m ∈ V /∂V , m ∈ Z ≥0 , in involution.
Example 2 A VA is commutative if [a λ b] = 0, for every a, b ∈ V . It follows
immediately from the definition that the category of commutative VAs is the same
as the category of differential algebras.
Remark 1 The (not necessarily commutative nor associative) product in a VA
corresponds to the normally ordered product of quantum fields in a CFT, while
the λ-bracket encodes the singular part of their operator product expansion (OPE).
We give a naive explanation of the latter sentence in a particular case. Consider the
VA λ-bracket of a Virasoro element u (recall Example 1 for its PVA analogue)
[u λ u] = (2λ + ∂)u +
c
12
λ
3 ,
where c ∈ C is called the central charge. Replace, in the above relation, u by a
quantum field, say T (w), ∂ by ∂ w and λ by ∂ w acting on the rational function
1
z−w .
Then we get
[T (w) ∂ w T (w)] →
1
z − w
=
∂ w T (w)
z − w
+
2T (w)
(z − w) 2 +
c/2
(z − w) 4 ,
which is the singular part of the OPE of the stress-energy tensor in CFT.
2.2 A Toy Model
The simplest example when all four objects in diagram (1) can be constructed is
obtained starting with a finite-dimensional Lie algebra g, with Lie bracket [· , ·],
and with a non-degenerate invariant symmetric bilinear form (· | ·).
The universal enveloping algebra of g, usually denoted by U(g), is an associative
algebra, and its classical limit is the symmetric algebra S(g), with the Kirillov–
Kostant Poisson bracket.
Furthermore, we have also a Lie conformal algebra Cur g = (F[∂] ⊗ g) ⊕ FK,
with the following λ-bracket:
[a λ b] = [a, b] + (a|b)Kλ , [a λ K] = 0 , for a, b ∈ g .
(3)
The universal enveloping vertex algebra of Cur g is the so-called universal affine
vertex algebra V (g), and its classical limit is the algebra of differential polynomials
V(g) = S(F[∂]g), with the PVA λ-bracket defined by (3). We refer to [14] for
the definition of the latter structures and the construction of the corresponding Zhu
maps. Thus, we get the following basic example of diagram (1):
185
and we have a representation of the Lie algebra V /∂V on V given by [
f, g] =
[f λ g]| λ=0 . A quantum integrable system consists in a collection of infinitely many
linearly independent elements
h m ∈ V /∂V , m ∈ Z ≥0 , in involution.
Example 2 A VA is commutative if [a λ b] = 0, for every a, b ∈ V . It follows
immediately from the definition that the category of commutative VAs is the same
as the category of differential algebras.
Remark 1 The (not necessarily commutative nor associative) product in a VA
corresponds to the normally ordered product of quantum fields in a CFT, while
the λ-bracket encodes the singular part of their operator product expansion (OPE).
We give a naive explanation of the latter sentence in a particular case. Consider the
VA λ-bracket of a Virasoro element u (recall Example 1 for its PVA analogue)
[u λ u] = (2λ + ∂)u +
c
12
λ
3 ,
where c ∈ C is called the central charge. Replace, in the above relation, u by a
quantum field, say T (w), ∂ by ∂ w and λ by ∂ w acting on the rational function
1
z−w .
Then we get
[T (w) ∂ w T (w)] →
1
z − w
=
∂ w T (w)
z − w
+
2T (w)
(z − w) 2 +
c/2
(z − w) 4 ,
which is the singular part of the OPE of the stress-energy tensor in CFT.
2.2 A Toy Model
The simplest example when all four objects in diagram (1) can be constructed is
obtained starting with a finite-dimensional Lie algebra g, with Lie bracket [· , ·],
and with a non-degenerate invariant symmetric bilinear form (· | ·).
The universal enveloping algebra of g, usually denoted by U(g), is an associative
algebra, and its classical limit is the symmetric algebra S(g), with the Kirillov–
Kostant Poisson bracket.
Furthermore, we have also a Lie conformal algebra Cur g = (F[∂] ⊗ g) ⊕ FK,
with the following λ-bracket:
[a λ b] = [a, b] + (a|b)Kλ , [a λ K] = 0 , for a, b ∈ g .
(3)
The universal enveloping vertex algebra of Cur g is the so-called universal affine
vertex algebra V (g), and its classical limit is the algebra of differential polynomials
V(g) = S(F[∂]g), with the PVA λ-bracket defined by (3). We refer to [14] for
the definition of the latter structures and the construction of the corresponding Zhu
maps. Thus, we get the following basic example of diagram (1):