18.276 Lecture Note 20
Lecture by Victor Kac, Scribed by Dongkwan Kim
April 30, 2015
Before we start, recall the following diagram presented on the first lecture.
Poisson vertex algebra o
Vertex algebra
Poisson algebra o
Associative algebra
For each of these structures, there is a notion of Hamiltonian reduction, called classical
Hamiltonian reduction on the left column and quantum Hamiltonian reduction on the right
column. In this lecture, we discuss the classical Hamiltonian reduction for Poisson vertex
algebras. Note that for a Poisson algebra it can be viewed as a special case.
20.1
Classical Hamiltonian Reduction of a Poisson Vertex Algebra
Let V be a Poisson vertex algebra, and suppose we are given a triple (V0 , I0 , ϕ) where V0
is a Poisson vertex algebra, I0 ⊂ V0 is a Poisson vertex algebra ideal, and ϕ : V0 → V is
a Poisson vertex algebra homomorphism. Then the Hamiltonian reduction associated to
(V0 , I0 , ϕ) is the differential algebra
W = W(V0 , I0 , ϕ) := (V/Vϕ(I0 ))adλ ϕ(V0 ) .
where Vϕ(I0 ) is the differential algebra ideal of V generated by ϕ(I0 ). We expect this to be
a well-defined Poisson vertex algebra, if we define the (obvious) λ-bracket as the following.
{f + Vϕ(I0 )λ g + Vϕ(I0 )} := {fλ g} + Vϕ(I0 )
Theorem 20.1. The λ-bracket on W is well-defined and endows the differential algebra
W with a structure of a Poisson vertex algebra.
Proof. We put W̃ = {f ∈ V | {ϕ(V0 )λ f } ⊂ Vϕ(I0 )[λ]} and show that this is a Poisson
vertex subalgebra of V. To this end, it suffices to check that W̃ is closed under derivation
and λ-bracket on V, that is for f, g ∈ W̃, ∂f ∈ W̃ and {fλ g} ∈ W̃[λ]. Indeed, we have
{ϕ(V0 )λ ∂f } ⊂ (λ + ∂){ϕ(V0 )λ f } ⊂ (λ + ∂)Vϕ(I0 )[λ] ⊂ Vϕ(I0 )[λ],
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since Vϕ(I0 ) is stable under derivation. Thus ∂f ∈ W̃. Also for any h ∈ V0 ,
{ϕ(h)λ {fµ g}} = {{ϕ(h)λ f }λ+µ g} + {fµ {ϕ(h)λ g}}
∈ {Vϕ(I0 )[λ]λ+µ g} + {fµ Vϕ(I0 )[λ]}
⊂ ({g−λ−µ−∂ Vϕ(I0 )} + {fµ Vϕ(I0 )})[λ]
⊂ Vϕ(I0 )[λ, µ]
using the Leibniz rule of the λ-bracket. Thus W̃ is a Poisson vertex subalgebra of V.
Now we claim Vϕ(I0 ) is an ideal of W̃. Indeed, for f ∈ Vϕ(I0 ), {ϕ(V0 )λ f } ⊂ Vϕ(I0 )[λ]
using the Leibniz rule since ϕ is a Poisson vertex algebra homomorphism and I0 ⊂ V0 is a
Poisson vertex algebra ideal of V0 . Thus we have Vϕ(I0 ) ⊂ W̃. Since Vϕ(I0 ) is a differential
algebra ideal of V, it only remains to check that for f ∈ W̃ we have {fλ ϕ(I0 )}, {ϕ(I0 )λ f } ⊂
Vϕ(I0 )[λ]. But it follows from the definition of W̃. (The other one comes from the skewsymmetry.)
Now we claim W = W̃/Vϕ(I0 ) as differential algebras. First of all, f + Vϕ(I0 ) ∈ W
for some f ∈ V if and only if {ϕ(V0 )λ f + Vϕ(I0 )} ⊂ Vϕ(I0 )[λ] if and only if {ϕ(V0 )λ f } ⊂
Vϕ(I0 )[λ] if and only if f ∈ W̃. Thus we have a natural homomorphism W → W̃/Vϕ(I0 ).
It is clearly injective, and the argument above shows that it is also surjective. Now we note
that the λ-bracket defined on W is nothing but that on W̃/Vϕ(I0 ) via this isomorphism,
whence the result since W̃/Vϕ(I0 ) is a priori a Poisson vertex algebra with respect to this
λ-bracket.
20.2
Quantum Hamiltonian Reduction of an Associative Algebra
Now we discuss quantum Hamiltonian reduction for an associative algebra. Let A be
an associative algebra, and (A0 , I0 , ϕ) be a triple such that A0 is an associative algebra,
I0 ⊂ A0 , and ϕ : A0 → A is an associative algebra homomorphism. Then we have the
following analogue of the previous theorem.
Theorem 20.2. W = W (A0 , I0 , ϕ) := (A/Aϕ(I0 ))ad ϕ(A0 ) is an associative algebra with
the well-defined product (a + Aϕ(I0 ))(b + Aϕ(I0 )) := ab + Aϕ(I0 ). Here Aϕ(I0 ) is a left
ideal generated by ϕ(I0 ).
Proof. (Exercise 20.1) We mimic the proof above and set W̃ = {f ∈ A | [ϕ(A0 ), f ] ⊂
Aϕ(I0 )}, where [a, b] = ab − ba. (Note that this satisfies the usual Jacobi identity by direct
calculation.) W̃ is an associative subalgebra of A since for b, c ∈ W̃ and a ∈ A0 ,
[ϕ(a), bc] = ϕ(a)bc − bcϕ(a)
= ϕ(a)bc − bϕ(a)c + bϕ(a)c − bcϕ(a)
= [ϕ(a), b]c + b[ϕ(a), c]
= [[ϕ(a), b], c] + c[ϕ(a), b] + b[ϕ(a), c].
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To show [[ϕ(a), b], c] ∈ Aϕ(I0 ), we use the following identity.
[xy, z] = xyz − zxy = xyz − xzy + xzy − zxy = x[y, z] + [x, z]y
The other terms are obviously in Aϕ(I0 ) from the definition of W̃ .
Meanwhile, ϕ(I0 ) ⊂ W̃ is clear since I0 ⊂ A0 is a two-sided ideal. Now we show
Aϕ(I0 ) ⊂ W̃ is also a two-sided ideal of W̃ . Indeed it suffices to show that for a ∈ I0 and
b ∈ W̃ we have ϕ(a)b ∈ Aϕ(I0 ). But it is true since ϕ(a)b = [ϕ(a), b] + bϕ(a) ∈ Aϕ(I0 ).
We now claim that W ' W̃ /Aϕ(I0 ). First of all, f + Aϕ(I0 ) ∈ W for some f ∈ A if
and only if [ϕ(A0 ), f + Aϕ(I0 )] ⊂ Aϕ(I0 ) if and only if [ϕ(A0 ), f ] ⊂ Aϕ(I0 ) if and only if
f ∈ W̃ . Thus we have a linear map W → W̃ /Aϕ(I0 ). This is clearly injective, and also
surjective by the argument above. Then we see that via this identification the product
on W is nothing but that of W̃ /Aϕ(I0 ), whence the result since W̃ /Aϕ(I0 ) is a priori an
associative algebra.
20.3
Construction of a Classical Affine W-Algebra
Let g be a finite dimensional Lie algebra with a non-degenerate invariant symmetric bilinear
form ( | ), and f ∈ g be a nilpotent element of g. Let V(g) = S(F[∂]g) (symmetric algebra)
and for a, b ∈ g define the λ-bracket
{aλ b}z := [a, b] + λ(a | b) − z(s | [a, b])
for a fixed s ∈ g and z ∈ F, which extends to V(g) by the Master formula. We also denote
{aλ b}H = [a, b] + λ(a | b),
{aλ b}K = (s | [a, b]).
We assume that we may embed sl2 = he, h, f i ,→ g such that f ∈ sl2 corresponds to the
original f ∈ g. This is always possible if g is reductive by the Jacobsom-Morozov theorem.
Then we have g = ⊕j∈ 1 Z gj , the eigenspace decomposition for 12 ad h. We let V0 = V(g>0 )
2
where g>0 = ⊕j∈ 1 Z>0 gj and ϕ : V0 → V be the inclusion. Also we define I0 ⊂ V0 to be a
2
differential algebra ideal generated by M = {m − (f |m) | m ∈ g≥1 }. Then by the argument
in the first section, we may construct a Poisson vertex algebra
W(g, f ) := W(V0 , I0 , ϕ).
Note that we have the induced family of λ-brackets {
Then we have the following theorem.
λ
}z = {
λ
}H − z{
λ
}K for z ∈ F.
Theorem 20.3. As a differential algebra, W(g, f ) is an algebra of differential polynomials
over gf , with explicitely known λ-brackets { λ }z .
Also, H and K on W(g, f ) are compatible, thus W(g, f ) is a bi-Poisson vertex algebra.
We want to know when the Lenard-Magri scheme of integrability works beginning with
ξ0 ∈ ker K. Here is a partial answer.
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Theorem 20.4. Let g be a simple finite dimensional Lie algebra. Then via the LenardMagri scheme, the system of partial differential equations corresponding to W(g, f ) is integrable provided that s = f + eθ is semisimple, where eθ is the highest root element.
P
For example, (Drinfeld-Sokolov) the case f = i eαi satisfies the theorem above. Meanwhile, when g = sl6 and f corresponds to the partition (4, 2) ` n, then it is not known
that the system of PDE is integrable or not. (This is the first open case.) Also, there is no
known case in which f + eθ is not semisimple but it is still integrable.
20.4
Non-Local Poisson Bracket
∂
Until now, we deal with the case {ui (x), uj (y)} = H(y, y 0 , · · · , y (n) ; ∂y
)δ(x − y) where H
is a matrix differential operator. We want to expand the argument to the case where H is
a pseudo-differential operator, e.g. H = ∂ −1 . Then the Poisson bracket is no longer local,
which we will call ”the non-local Poisson bracket.” Indeed, in this case {fλ g} ∈ V ((λ−1 )),
the formal Laurent series in λ−1 with coefficient in V .
In this setting, the Master formula is the same. Meanwhile,
the skew-symmetry involves
P
i ∂ i , thus we can still have the
(−1)
(∂ +λ)−1 , which can be regarded as λ(1+ λ∂ )−1 = λ ∞
i=0
λi
corresponding skew-symmetry formula. For Jacobi identity, we assume that the λ-bracket
is admissible, which is defined by
{{aλ b}µ c} ∈ V [[λ−1 , µ−1 , (λ + µ)−1 ]][λ, µ].
For example, if H(∂) = A(∂)/B(∂) for some differential operators A(∂) and B(∂), then
the corresponding λ-bracket is admissible.
20.5
Dirac Reduction of a Poisson Vertex Algebra
Let V be a Poisson vertex algebra. Suppose given θ1 , · · · , θm ∈ V (constraints) such that
C(∂) is an invertible matrix differential operator where C(λ) := ({θβ λ θα })m
α,β=1 . In this
case we have the Dirac modification of the λ-bracket as the following.
m
X
{fλ g}D := {fλ g} −
{θβ λ+∂ g}→ (C −1 )β,α (λ + ∂) {fλ θα }
α,β=1
We claim that this is a well-defined λ-bracket. Here we only check the sesquilinearity,
skew-symmetry and that θα are central. (Exercise 20.2) For f, g ∈ V, we have
D
{∂f λ g} = {∂f λ g} −
m
X
{θβ λ+∂ g}→ (C −1 )β,α (λ + ∂) {∂f λ θα }
α,β=1

= −λ {fλ g} −
m
X
{θβ λ+∂ g}→
α,β=1
= −λ{fλ g}D ,
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
(C −1 )β,α (λ + ∂) {fλ θα }
D
{fλ ∂g} = {fλ ∂g} −
m
X
{θβ λ+∂ ∂g}→ (C −1 )β,α (λ + ∂) {fλ θα }
α,β=1

m
X
= (λ + ∂) {fλ g} −

{θβ λ+∂ g}→ (C −1 )β,α (λ + ∂) {fλ θα }
α,β=1
D
= (λ + ∂){fλ g} ,
thus the sesquilinearity is clear.
For skew-symmetry, first we investigate the skew-adjointness of C(∂). To that end, we
recall the skew-symmetry of the original λ-bracket {θαλ θβ } = −{θβ −λ−∂ θα }. By considering this term as a symbol of C(∂), we see that C(∂) is skew-adjoint, i.e. C ∗ (∂) = −t C(∂).
Thus C −1 (∂) also has the same
property. ToPbe precise, if we write (C −1 )α,β (λ) =
P
P
n
n
−1 )
n
n
n n
β,α (λ) =
n dβ,α λ = −
n (−λ − ∂) dα,β .
n dα,β λ , then (C
Now we let ∂1 , ∂2 , ∂3 be differential operators acting only on {θβ λ g}, the coefficient of
C −1 , and {fλ θα }, respectively. Then we may write
{θβ λ+∂ g}→ (C −1 )β,α (λ + ∂) {fλ θα } = {θβ λ+∂2 +∂3 g} (C −1 )β,α (λ + ∂3 ) {fλ θα },
which is by skew-symmetry and the skew-adjointness of C the same as
−{g−λ−∂1 −∂2 −∂3 θβ }((C −1 )α,β (−λ − ∂2 − ∂3 )){θα−λ−∂3 f }.
Here the meaning of (C −1 )α,β (−λ − ∂2 − ∂3 ) is clear; ∂2 is regarded as positioned on the left
of the coefficients of (C −1 )α,β . Meanwhile, if we substitute
λ with −λ−∂ = −λ−∂1 −∂2 −∂3
−1
and change f and g, then {θβ λ+∂ g}→ (C )β,α (λ + ∂) {fλ θα } becomes
{θβ −λ−∂1 f } (C −1 )β,α (−λ − ∂1 − ∂2 ) {g−λ−∂1 −∂2 −∂3 θα }
Also the meaning of each term is clear in this expression. But by changing α and β, we easily
see that this expression is the additive inverse of {θβ λ+∂ g}→ (C −1 )β,α (λ + ∂) {fλ θα }.
Now along with this fact and the skew-symmetry of the original λ-bracket, the skewsymmetry of { λ }D follows.
It remains to show that θi are central. By definition of C −1 (∂), we have
δβ,γ =
m
X
(C
−1
)β,α (λ + ∂)Cα,γ (λ) =
α=1
m
X
(C −1 )β,α (λ + ∂){θγ λ θα }.
α=1
Thus for any θγ we have
{θγ λ f }D = {θγ λ f } −
m
X
{θβ λ+∂ f }→ (C −1 )β,α (λ + ∂) {θγ λ θα }
α,β=1
= {θγ λ f } − {θγ λ+∂ f }→ 1 = 0.
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But this is what we want to prove. Note that {fλ θγ }D = 0 follows by skew-symmetry.
Likewise, one can check the other conditions for λ-bracket, i.e. Jacobi identity, Leibniz
rules, etc. Thus we see that V/hθ1 , · · · , θm i with the Dirac modified λ-bracket is a non-local
Poisson vertex algebra. (Here hθ1 , · · · , θm i is the differential ideal generated by θ1 , · · · , θm .)
Example 20.5. We keep the notation on the previous section and take V = V(sln ) =
F[e, h, f, e0 , h0 , f 0 , · · · ]. Then for s = h, h is central with respect to the bracket { λ }K ,
0
0 0
thus V red = V/hh,
h , · ·· i = F[e, f, e , f , · · · ] is a Poisson vertex algebra with the Poisson
0 −1
structure K =
. But h is not central for { λ }H . Thus we modify this λ-bracket
1 0
using the procedure above with h = θ to have { λ }D , and it corresponds to
1 0
f ∂ −1 ◦ f −f ∂ −1 ◦ e
D
H =∂
+
.
0 1
−e∂ −1 ◦ f e∂ −1 ◦ e
Note that h becomes central with respect to { λ }D
H . Then we get the first nontrivial
system, which corresponds to the nonlinear Schorödinger equation
de
1
= e00 − e2 f,
dt
2
df
1
= − f 00 + ef 2 .
dt
2
References
[1] Drinfel’d, V. G., and V. V. Sokolov. ”Lie algebras and equations of Korteweg-de Vries
type.” Journal of Soviet mathematics 30.2 (1985): 1975-2036.
[2] De Sole, Alberto, Victor G. Kac, and Daniele Valeri. ”Classical W-Algebras and Generalized Drinfeld-Sokolov Bi-Hamiltonian Systems Within the Theory of Poisson Vertex
Algebras.” Communications in Mathematical Physics 323.2 (2013): 663-711.
[3] De Sole, Alberto, and Victor G. Kac. ”Non-local Poisson structures and applications
to the theory of integrable systems.” Japanese Journal of Mathematics 8.2 (2013):
233-347.
[4] De Sole, Alberto, Victor G. Kac, and Daniele Valeri. ”Dirac reduction for Poisson vertex
algebras.” Communications in Mathematical Physics 331.3 (2014): 1155-1190.
[5] Dirac, Paul Adrien Maurice. ”Generalized hamiltonian dynamics.” Can. J. Math 2.2
(1950): 129-148.
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