Symplectic groupoids for cluster Poisson
structures
Songhao Li
University of Notre Dame
Gone Fishing Poisson geometry meeting 2017
May, 2017
Based on joint work with Dylan Rupel
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Outline
Cluster Poisson structure of A2
Poisson spray and symplectic groupoid
Poisson spray on the pentagon
Averaged symplectic form and Moser trick
Cluster symplectic groupoid
General cluster Poisson structures
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Cluster Poisson structure of A2
Consider the pentagon P as a manifold with corners.
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Cluster Poisson structure of A2
Consider the pentagon P as a manifold with corners.
x2 = 0
x1 = 0
x3 = 0
x5 = 0
x4 = 0
We take five coordinate charts: xi ≥ 0
(x1 , x2 ),
(x2 , x3 ),
(x3 , x4 ),
(x4 , x5 ),
(x5 , x1 ).
with the coordinate transformations as follows:
1 + x2
,
x3 =
x1
1 + x3
1 + x1 + x2
x4 =
=
x2
x1 x2
1 + x4
1 + x1
x5 =
=
x3
x2
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Cluster Poisson structure of A2
In cluster algebra language, the charts (xi , xi+1 ) are the clusters of the
cluster algebra for the Dynkin diagram A2 . The pentagon is the
corresponding totally positive cluster manifold. The coordinate
transformations
(xi , xi+1 ) 7→ (xi+1 , xi+2 ),
xi+1 = xi+1 ,
xi+2 =
1 + xi+1
xi
are called the cluster mutations.
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Cluster Poisson structure of A2
In cluster algebra language, the charts (xi , xi+1 ) are the clusters of the
cluster algebra for the Dynkin diagram A2 . The pentagon is the
corresponding totally positive cluster manifold. The coordinate
transformations
(xi , xi+1 ) 7→ (xi+1 , xi+2 ),
xi+1 = xi+1 ,
xi+2 =
1 + xi+1
xi
are called the cluster mutations.
There is a log canonical Poisson structure {xi , xi+1 } = xi xi+1 on the
∂
∧ ∂x∂i+1 . The Poisson
pentagon P, or equivalently π = xi xi+1 ∂x
i
structure π is non-degenerate in the interior of P, and is zero on the
boundary of P.
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Cluster Poisson structure of A2
In cluster algebra language, the charts (xi , xi+1 ) are the clusters of the
cluster algebra for the Dynkin diagram A2 . The pentagon is the
corresponding totally positive cluster manifold. The coordinate
transformations
(xi , xi+1 ) 7→ (xi+1 , xi+2 ),
xi+1 = xi+1 ,
xi+2 =
1 + xi+1
xi
are called the cluster mutations.
There is a log canonical Poisson structure {xi , xi+1 } = xi xi+1 on the
∂
∧ ∂x∂i+1 . The Poisson
pentagon P, or equivalently π = xi xi+1 ∂x
i
structure π is non-degenerate in the interior of P, and is zero on the
boundary of P.
Remark
If the xi ’s are holomorphic coordinates, i.e. xi ∈ C∗ , then the
compactification of the cluster manifold is the del Pezzo surface of
degree 5.
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Poisson spray and symplectic groupoid
Poisson spray
For a Poisson manifold (M, π), a vector field X ∈ X(T ∗ M) is a Poisson
spray if
1. for (x, p) ∈ T ∗ M,
(τM )∗ (X )(x,p) = π(p)
where (τM ) : T ∗ M → M is the projection;
2. X is homogeneous of degree 1, i.e.
(mλ )∗ (X ) = λX
where mλ is the fiberwise scaling map
mλ : T ∗ M → T ∗ M,
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(x, p) 7→ (x, λp).
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Poisson spray and symplectic groupoid
Crainic-Marcut 2011, Cabrera-Marcut-Salazar Preprint
For a Poisson manifold (M, π) with a Poisson spray X ∈ X(T ∗ M), a
neighbourhood U of the zero section of T ∗ M is a local symplectic
groupoid over (M, tπ) with the following structure:
1. the source map α = τM : U → M is the bundle projection ;
2. the target map is
β = τM ◦ ϕtX
β : U → M,
where ϕtX : T ∗ M → T ∗ M is the time-t-flow of the Poisson spray X ;
3. the multiplication is the concatenation of the flow of X ; and
4. the symplectic form on U is
1
ωt =
t
Z
0
t
(ϕsX )∗ ω0 ds.
where ω0 is canonical symplectic form on T ∗ M.
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Poisson spray on the pentagon
There are two possible obstructions to extending the local symplectic
groupoid U to all of T ∗ M:
1. The Poisson spray X may not be complete, or its flow may have
loops.
2. The averaged 2-form ω t may be degenerate.
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Poisson spray on the pentagon
There are two possible obstructions to extending the local symplectic
groupoid U to all of T ∗ M:
1. The Poisson spray X may not be complete, or its flow may have
loops.
2. The averaged 2-form ω t may be degenerate.
Back to the Poisson pentagon (P, π), in the initial cluster (x1 , x2 ),
where π = x1 x2 ∂x∂ 1 ∧ ∂x∂ 2 we choose the Poisson spray:
X = −x2 p2 x1
∂
∂
∂
∂
+ x1 p1 x2
− x1 p1 p2
+ x2 p2 p1
.
∂x1
∂x2
∂p2
∂p1
The flow of X is complete and has no loops:
ϕtX : (p1 , p2 , x1 , x2 ) 7→ (etx2 p2 p1 , e−tx1 p1 p2 , e−tx2 p2 x1 , etx1 p1 x2 )
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Poisson spray on the pentagon
What about the other clusters? In the next cluster (x2 , x3 , p2 , p3 ) where
x2 , x3 ≥ 0, the flow of X is again complete and has no loops:
ϕtX :(p2 , p3 , x2 , x3 )
7→ e−tx1 p1 p2 +
e−tx2 p2 x12 p1 tx1 p1
x1 p1
1 + etx1 p1 x2
x
,
,
−
,
e
2
1 + etx1 p1 x2 1 + etx1 p1 x2
e−tx2 p2 x1
In the same way, one can check that the flow of X is complete and has
no loops on all clusters.
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Poisson spray on the pentagon
What about the other clusters? In the next cluster (x2 , x3 , p2 , p3 ) where
x2 , x3 ≥ 0, the flow of X is again complete and has no loops:
ϕtX :(p2 , p3 , x2 , x3 )
7→ e−tx1 p1 p2 +
e−tx2 p2 x12 p1 tx1 p1
x1 p1
1 + etx1 p1 x2
x
,
,
−
,
e
2
1 + etx1 p1 x2 1 + etx1 p1 x2
e−tx2 p2 x1
In the same way, one can check that the flow of X is complete and has
no loops on all clusters.
Total positivity
In general, the reason that the Poisson spray X is complete is the total
positivity for cluster algebras.
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Averaged symplectic form and Moser trick
Next, we check the averaged 2-form. In the chart (x1 , x2 ), we have
Z
1 t s ∗
(ϕ ) ω0 ds
t 0 X
= dp1 dx1 + dp2 dx2
ωt =
− t (p1 p2 dx1 dx2 + p1 x2 dx1 dp2 + x1 p2 dp1 dx2 + x1 x2 dp1 dp2 ) ,
ω t ∧ ω t = 2dp1 dx1 dp2 dx2 .
So ω t is non-degenerate.
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Averaged symplectic form and Moser trick
Next, we check the averaged 2-form. In the chart (x1 , x2 ), we have
Z
1 t s ∗
(ϕ ) ω0 ds
t 0 X
= dp1 dx1 + dp2 dx2
ωt =
− t (p1 p2 dx1 dx2 + p1 x2 dx1 dp2 + x1 p2 dp1 dx2 + x1 x2 dp1 dp2 ) ,
ω t ∧ ω t = 2dp1 dx1 dp2 dx2 .
So ω t is non-degenerate.
In fact, we have a 1-parameter family of exact symplectic structures
ω t = dθt , where
Z
1 t s ∗
θt =
(ϕ ) θ0 ds
t 0 X
and θ0 is the tautological 1-form on T ∗ P.
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Averaged symplectic form and Moser trick
Moser trick
d
The time-dependent vector field Xt defined by ιXt ω t = dt
θt is 12 X .
That is, Xt is half of the Poisson spray X , which is complete.
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Averaged symplectic form and Moser trick
Moser trick
d
The time-dependent vector field Xt defined by ιXt ω t = dt
θt is 12 X .
That is, Xt is half of the Poisson spray X , which is complete.
Conclusion
For the Poisson pentagon (P, π), if we choose the Poisson spray
X = −x2 p2 x1
∂
∂
∂
∂
+ x1 p1 x2
− x1 p1 p2
+ x2 p2 p1
,
∂x1
∂x2
∂p2
∂p1
then we have a 1-parameter family of symplectic groupoids
(T ∗ P, ω t ) ⇒ (P, tπ) and
1 ∗
t
ω t = ϕX2
ω0 .
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Cluster symplectic groupoid
We could take an alternative point of view, akin to cluster algebras. On
each cluster (pi , pi+1 , xi , xi+1 ), we choose the Poisson spray:
Xi = −xi+1 pi+1 xi
∂
∂
∂
∂
+ xi pi xi+1
− xi pi pi+1
+ xi+1 pi+1 pi
,
∂xi
∂xi+1
∂pi+1
∂pi
then the symplectic groupoid structure on this cluster is given by
α : (pi , pi+1 , xi , xi+1 ) 7→ (xi , xi+1 )
β : (pi , pi+1 , xi , xi+1 ) 7→ (e−xi+1 pi+1 xi , exi pi xi+1 )
0
m : (pi , pi+1 , xi , xi+1 ), (pi0 , pi+1
, e−xi+1 pi+1 xi , exi pi xi+1 )
0
+ pi+1 , xi , xi+1
7→ e−xi+1 pi+1 pi0 + pi , exi pi pi+1
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Cluster symplectic groupoid
We could take an alternative point of view, akin to cluster algebras. On
each cluster (pi , pi+1 , xi , xi+1 ), we choose the Poisson spray:
Xi = −xi+1 pi+1 xi
∂
∂
∂
∂
+ xi pi xi+1
− xi pi pi+1
+ xi+1 pi+1 pi
,
∂xi
∂xi+1
∂pi+1
∂pi
then the symplectic groupoid structure on this cluster is given by
α : (pi , pi+1 , xi , xi+1 ) 7→ (xi , xi+1 )
β : (pi , pi+1 , xi , xi+1 ) 7→ (e−xi+1 pi+1 xi , exi pi xi+1 )
0
m : (pi , pi+1 , xi , xi+1 ), (pi0 , pi+1
, e−xi+1 pi+1 xi , exi pi xi+1 )
0
+ pi+1 , xi , xi+1
7→ e−xi+1 pi+1 pi0 + pi , exi pi pi+1
Remark
Here, we do not have a global Poisson spray on the pentagon P.
Instead we have a different Poisson spray Xi on each cluster, and
therefore a different symplectic groupoid structure on each cluster.
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Cluster symplectic groupoid
The cluster mutation
(xi , xi+1 ) 7→ (xi+1 , xi+2 ),
xi+1 = xi+1 ,
xi+2 =
1 + xi+1
xi
lifts the cluster mutation on groupoid
0
0
(pi , pi+1
, xi , xi+1 ) 7→ (pi+1 , pi+2
, xi+1 , xi+2 )
1 + xi+1
,
xi+1 = xi+1 ,
xi+2 =
xi
x 2 pi
0
1 + exi pi xi+1
0
pi+2
=− i
,
exi+1 pi+1 = exi+1 pi+1
.
1 + xi+1
1 + xi+1
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Cluster symplectic groupoid
The cluster mutation
(xi , xi+1 ) 7→ (xi+1 , xi+2 ),
xi+1 = xi+1 ,
xi+2 =
1 + xi+1
xi
lifts the cluster mutation on groupoid
0
0
(pi , pi+1
, xi , xi+1 ) 7→ (pi+1 , pi+2
, xi+1 , xi+2 )
1 + xi+1
,
xi+1 = xi+1 ,
xi+2 =
xi
x 2 pi
0
1 + exi pi xi+1
0
pi+2
=− i
,
exi+1 pi+1 = exi+1 pi+1
.
1 + xi+1
1 + xi+1
Remark
I
The groupoid cluster mutations also has a periodicity of 5.
I
This works for holomorphic coordinates as well.
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General cluster Poisson structures
General cluster algebras
To each cluster algebra A of rank n, there is a totally positive cluster
Poisson manifold PA . If A is finite type, i.e. A has finite many clusters,
then PA is a polyhedron, called the associahedron of A.
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General cluster Poisson structures
General cluster algebras
To each cluster algebra A of rank n, there is a totally positive cluster
Poisson manifold PA . If A is finite type, i.e. A has finite many clusters,
then PA is a polyhedron, called the associahedron of A.
PA has the log canonical Poisson structure:
π=
X
0≤i,j≤n
Bij xi xj
∂
∂
∧
.
∂xi ∂xj
where Bij is the ‘inverse’ of the exchange matrix of A. Each facet of PA
is a Poisson submanifold.
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General cluster Poisson structures
In the initial cluster, we choose the Poisson spray:
X =
X
0≤i,j≤n
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Bij xi xj pi
X
∂
∂
−
Bij pi pj xi
.
∂xj
∂pj
0≤i,j≤n
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General cluster Poisson structures
In the initial cluster, we choose the Poisson spray:
X =
X
0≤i,j≤n
Bij xi xj pi
X
∂
∂
−
Bij pi pj xi
.
∂xj
∂pj
0≤i,j≤n
Conclusion
The Poisson spray X extends to PA and is complete without loops. We
have a 1-parameter family of symplectic groupoids
(T ∗ PA , ω t ) ⇒ (P, tπ) and
1 ∗
t
ω t = ϕX2
ω0 .
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Thank you!
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