Poisson Geometry of Noncommutative Cluster …One deﬁnes an algebra of polyvector ﬁelds as D = T C1(M)D 1. Puzzle: Der(A;A) is no longer an A-module for a noncommutative algebra

Poisson Geometry of Noncommutative ClusterAlgebras

Semeon Arthamonov

UC Berkeley

November 11, 2019

Canonical Bases, Cluster Structures and NC Birational Geometry

AMS Fall Western Sectional Meeting, UC Riverside

S. Arthamonov Poisson Geometry of NC Cluster Algebras AMS Sectional Meeting, UC Riverside 1 / 32

Cluster algebras associated to triangulated surfaces

Figure: Flip of an ideal triangulation corresponds to mutation.

M. Gekhtman , M. Shapiro, and A. Vainshtein Cluster algebras andWeil-Petersson forms. Duke Mathematical Journal, 127(2), 291-311, 2005.

V. Fock, A. Goncharov, A. Moduli spaces of local systems and higher Teichmullertheory. Publications Mathematiques de l’IHES, 103, 1-211, 2006.

S. Fomin, M. Shapiro, and D. Thurston. Cluster algebras and triangulatedsurfaces. Part I: Cluster complexes. Acta Mathematica, 201(1):83–146, 2007.

A. Goncharov, and R. Kenyon. Dimers and cluster integrable systems. Annalesscientifiques de l’Ecole Normale Superieure. Vol. 46. No. 5., 2013.


Ribbon Graphs

DefinitionA ribbon graph Γ is a graph with cyclic order of edges adjacent to each vertex.

(a) Ribbon Graph(b) Disc in SΓ corresponding tothe vertex.

Figure: Surface with boundary SΓ associated to a ribbon graph.

Each ribbon graph Γ defines an oriented surface SΓ with boundary.


Ideal triangulations and bipartite graphs

Figure: Bipartite ribbon graph associated to triangulation of surface Σ.

DefinitionA conjugate surface SΓ associated to the bipartite ribbon graph Γ is a surfacecorresponding to the ribbon graph with reversed cyclic order of edges at eachblack vertex.

Surface SΓ has the same fundamental group as the underlying graph

π1(SΓ) = π1(Γ). (1)


Graph connections and edge weights

Fix arbitrary orientation of Γ and let G be a linear algebraic group.

DefinitionGraph connection is an assignment of a parallel transport ge ∈ G to eachoriented edge e ∈ E(Γ).

The gauge group GV (Γ) = {V (Γ)→ G, v 7→ hv} acts on a graph connections

ge 7→ ht(e)geh−1s(e),

where s(e) and t(e) stand for the source and target of an edge e respectively.

Conjugacy class of a parallel transport MF = gek . . . ge2ge1

along the edge loop is invariant under the action of GV (Γ).MF

Variables of a cluster chart in [G.-K.] correspond to G = GL(1).


Cluster algebras and Poisson Geometry

Let k be a ground field of characteristic zero.

DefinitionLet A be a commutative associative algebra over k, a k-linear map

{, } : A⊗A −→ A

is called a Poisson bracket on A if it satisfies for all f ,g,h ∈M

{f ,g} = −{g, f} skew-symmetry condition,{f ,gh} = g{f ,h}+ {f ,g}h Leibnitz identity,{f , {g,h}}+ {g, {h, f}}+ {h, {f ,g}} = 0 Jacobi identity,

Geometric Cluster Algebras can be equipped with a Poisson bracketcompatible with mutations.

M. Gekhtman, M. Shapiro, and A. Vainshtein. Cluster algebras and Poissongeometry. Moscow Mathematical Journal, 3(3):899–934, 2003.


Poisson bracket on graph connectionsEach 1-dimensional representation

ϕ ∈ Hom(π1(SΓ),C×)

is determined by

y1 = ϕ(M1), . . . , yn = ϕ(Mn).

We can equip C[y1, . . . , yn] with a Poisson bracket as follows

{yi , yj} =∑

p

εi,j (p)yiyj ,

εij (p) =

+1

Mj Mi

p

−1Mi Mj

p

M1 M2

M3


Rectangle move

y1

y2 y3

y4

y0

z1

z2 z3

z4

z0

Figure: Rectangle move in one dimensional case.

Proposition (Goncharov-Kenyon’2013)The following map extends to a homomorphism of Poisson algebras

τ :

z0 → y−10 ,

zi → yi (1 + y0), i = 1,3,zi → yi (1 + y−1

0 )−1, i = 2,4.


Noncommutative cluster algebras

Cluster algebras with noncommutative edge weightsA. Berenstein, V. Retakh Noncommutative marked surfaces. Advances inMathematics, 328, 1010-1087, 2018.

Can we equip them with a “Poisson bracket” compatible with mutations?Spoiler: Yes, but there are several challenges:

1 Usual notion of a Poisson bracket is too restrictive. In everynoncommutative ring it forced to be a multiple of the commutator.1

2 Can no longer use loops without a base point to define variables of acluster chart.

3 Considering loops with base point will destroy “locality” of the mutation.

Solution: Introduce a noncommutative bi-vector field acting on open arcs.

1for a proof, see D. Farkas and G. Letzter. Ring theory from symplectic geometry. Journal ofPure and Applied Algebra, 125(1–3):155 – 190, 1998.


Algebra of polyvector fields

A vector field d onM can be viewed as the derivation of C∞(M), the algebraof smooth functions onM

d : C∞(M)→ C∞(M), d(fg) = fd(g) + d(f )g,

for all f ,g ∈ C∞(M).

LemmaThe space of vector fields

D1 = Der(C∞(M),C∞(M))

forms a C∞(M)-module.

One defines an algebra of polyvector fields as D• = TC∞(M)D1.

Puzzle: Der(A,A) is no longer an A-module for a noncommutative algebra A.


Double Geometry

W. Crawley-Boevey, P. Etingof, and V. Ginzburg. Noncommutative geometry andquiver algebras. Advances in Mathematics, 209(1):274 – 336, 2007

M. Van den Bergh. Double Poisson algebras. Trans. Amer. Math. Soc.,360:5711–5769, 2008.

DefinitionLet A be an associative algebra. We say that map δ is a noncommutativevector field if

δ : A → A⊗A, δ(ab) = (a⊗ 1)δ(b) + δ(a)(1⊗ b)

for all a,b ∈ A.

LemmaNoncommutative vector fields DA = Der(A,A⊗A) form an A-bimodule.

One defines a noncommutative algebra of polyvector fields as TADA


Double derivations of a ring with many objects

DefinitionLet C be a small k-linear category. For all V ,W ∈ Obj C we say that a map

δ : Mor C → hom(W ,−)⊗ hom(−,V )

is a (V ,W )-vector field if

δ(f ◦ g) = (f ⊗ 1V ) ◦ δ(g) + δ(f ) ◦ (1W ⊗ g).

for all composable f ,g ∈ Mor C. Here 1V and 1W are the identity morphismson V and W .

In what follows we denote the space of (V ,W )-vector fields as D1V ,W . Let

(a ? δ ? b)(f ) = (δ′(f ) ◦ b)⊗ (a ◦ δ′′(f )) (3)

LemmaD1 is a covariant functor on C × Cop.


Modules over a ring with many objects

Fix a small k-linear category C.

Definition (Tensor product)Let R be a contrvariant functor on C and L be a covariant functor on C, thetensor product R ⊗C L is defined as

R ⊗C L =

⊕V∈Obj C

RV ⊗ LV

/ρ◦f⊗λ∼ρ⊗f◦λ

.

Definition (Trace)LetM be a bifunctor on C, the trace over C is defined as

trC : M→M\ :=

⊕X∈Obj C

MX ,X

/f◦m∼m◦f

.


Polyvector fieldsThe space of k-vector fields associated to V ,W ∈ Obj C is defined as

DkV ,W =

⊕U1,...,Uk−1∈Obj C

D1V ,U1⊗C . . . ⊗C D1

Uk−1,W ,

where for k = 0 we assume that

D0V ,W = hom(W ,V ).

CorollaryDk is a covariant functor on C × Cop.

D•V ,W =∞⊕

k=0

DkV ,W

We define the category V of polyvector fields on C as

Obj V = Obj C, homV(W ,V ) = D•V ,W .


Traces of polyvector fields and polyderivationsLet δ1, . . . , δk be a chain of composable vector fields. The tracetrC(δ1 ? · · · ? δk ) is equivalent to the following map

trC(δ1 ? · · · ? δk ) : (Mor C)⊗k → (Mor C)⊗k ,

f1 ⊗ · · · ⊗ fk 7→ (δ′k (fk ) ◦ δ′′1 (f1))⊗ (δ′1(f1) ◦ δ′′2 (f2))⊗ . . .

· · · ⊗ (δ′k−1(fk−1) ◦ δ′′k (fk )).

Proposition∆ = trC(δ1 ? · · · ? δk ) is a polyderivation, i.e.,

∆(h1⊗ · · · ⊗ f ◦ g↑j

⊗ · · · ⊗ hk )

=(1t(hk ) ⊗ · · · ⊗ f↑

j+1

⊗ · · · ⊗ 1t(hk−1)) ◦∆(h1 ⊗ · · · ⊗ g↑j

⊗ · · · ⊗ hk )

+ ∆(h1 ⊗ · · · ⊗ f↑j

⊗ · · · ⊗ hk ) ◦ (1s(h1) ⊗ · · · ⊗ g↑j

⊗ · · · ⊗ 1s(hk ))


Double Quasi Poisson Bracket on a category

DefinitionA k-linear map R is said to be a Double Quasi Poisson Bracket if it satisfies

Skew-Symmetry condition

R(f ⊗ g) = −(R(g ⊗ f )

)op

Double Leibnitz Identity

R((f ◦ g)⊗ h) =(1t(h) ⊗ f ) ◦ R(g ⊗ h) + R(f ⊗ h) ◦ (g ⊗ 1s(h)),

R(f ⊗ (g ◦ h)) =(g ⊗ 1t(f )) ◦ R(f ⊗ h) + R(f ⊗ g) ◦ (1s(f ) ⊗ h).

Double Quasi Jacobi Identity

R1,2 ◦ R2,3 + R2,3 ◦ R3,1 + R3,1 ◦ R1,2 =∑

V∈Obj C

trC(∂V ? ∂V ? ∂V ).

Ri,j (f1 ⊗ · · · ⊗ fn) =f1 ⊗ · · · ⊗ R′(fi ⊗ fj )︸︷︷︸i

⊗ · · · ⊗ R′′(fi ⊗ fj )︸︷︷︸j

⊗ · · · ⊗ fn.


Category associated to a ribbon graph

C0 = kπ1(SΓ,V1, . . . ,Vn).

The objects Obj C0 = {Vi} correspond to marked points.

x3

x1x2

(a) Disk corresponding to white vertex

f1f2

f−11 ◦ f−1

2

(b) Disc corresponding to black vertex

Figure: Building blocks for bipartite graph with trivalent black vertices


Double Quasi Poisson BivectorFor each object V ∈ C0 consider a noncommutative bivector

PV =12

∑i<j

(xj ?

∂

∂xi? xi ?

∂

∂xj− xi ?

∂

∂xj? xj ?

∂

∂xi

).

Here ∂∂fi∈ Ds(fi ),t(fi ) is a vector field on defined on generators of C0 as

∂

∂fi(fj ) =

{1t(fi ) ⊗ 1s(fi ), i = j ,0, i 6= j .

LemmaThe following map is a double Quasi Poisson Bracket on C0

{{, }}=∑

V∈Obj C0

trC0PV .

G. Massuyeau and V. Turaev. Quasi-Poisson structures on representation spacesof surfaces. IMRN, 2014(1):1–64, 2012.


Noncommutative rectangle move

Let Csub1 ⊂ C1 be a subcategory generated by Y±1

1 ,Y±12 ,Y±1

3 ,Y±14 . Similarly

we define a subcategory Csub2 ⊂ C2.

Y1

Y2 Y3

Y4

v1

v2

v3

v4

(a) Original morphisms

Z1

Z2 Z3

Z4

v1

v2

v3

v4

(b) Morphisms after the move

Figure: Rectangle move


Quasi Poisson Functor

Now let τ : C2 → C1 be a functor defined as

τ(Z1) =Y1 ◦ f1(M), τ(Z4) =f4(M) ◦ Y4,

τ(Z2) =Y2 ◦ Y1 ◦ f2(M) ◦ Y−11 , τ(Z3) =Y−1

4 ◦ f3(M) ◦ Y4 ◦ Y3,

where f1, . . . , f4 are the same as in one-dimensional case:

f1(M) = f3(M) = (1v1 + M)−1, f2(M) = f4(M) = 1v1 + M−1.

τ(Zi ) = Yi i ≥ 5.

Theorem (S.A.’2017)The functor τ preserves Double Quasi Poisson Bracket:

τ({{Zi ⊗ Zj}}

)= {{τ(Zi )⊗ τ(Zj )}}, 1 ≤ i , j ≤ n.


Representation SchemeFollowing general philosophy by M. Kontsevich any algebraic property thatmakes geometric sense is mapped to its commutative counterpart by

Representation Functor

RepN : fin. gen. Associative algebras→ Affine schemes,RepN(A) = Hom(A,MatN(C)).

ϕ(x (i)) =

x (i)

11 . . . x (i)1N

......

x (i)N1 . . . x (i)

NN

. (5)

Ring C[V] := C[x (i)

j,k

]/ϕ(R) defines Representation variety. The invariant

subring C[V]GLN ⊂ C[V] defines a Character Variety.2

Maxim Kontsevich. Formal (non)-commutative symplectic geometry. The GelfandMathematical Seminars, 1990–1992, pages 173–187. Birkhauser Boston, 1993.

2Both are not actually varieties, in general, but rather affine schemes.


Induced Brackets on Representation Scheme

Let {{, }} be a double Quasi Poisson bracket. Define induced bracket {, }V ongenerators of C[V] as{

x (m)ij , x (n)

kl

}V= ϕ

({{x (m) ⊗ x (n)}}

)(kj),(il)

(6)

and then extend it to the entire C[V]⊗ C[V] by Leibnitz identities

{ab, c}V =a{b, c}V + b{a, c}V , (7)

{a,bc}V =c{a,b}V + b{a, c}V . (8)

Lemma{, }V is a Quasi-Poisson bracket.

1 A. Alekseev, Y. Kosmann-Schwarzbach, and E. Meinrenken. Quasi-Poissonmanifolds. Canad. J. Math., (54):3–29, 2002.


Induced Poisson Bracket

PropositionThe following restriction

{ , }V : C[V]GLN ⊗ C[V]→ C[V] (10)

satisfies the left Loday-Jacobi identity: for all f ,g ∈ C[V]GLN and h ∈ C[V] :

{f , {g,h}V}V − {g, {f ,h}V}V = {{f ,g}V ,h}V . (11)

For all f ,g ∈ C[V]GLN we have {f ,g}V ∈ C[V]GLN and {f ,g}V = −{g, f}V .

PropositionThe following restriction of {, }V

{, }inv : C[V]GLN ⊗ C[V]GLN → C[V]GLN (13)

is a Poisson bracket.


Example 1: Kontsevich system

abc

(a) Original ribbon graph

a

b

c

v

u

(b) Conjugate surface T\D

Figure: Conjugate surface for Kronecker quiver with three vertices

Here CK = k〈u±1, v±1〉 becomes a group algebra of a π1(T\D).


Bracket on a torus and Kontsevich mapBracket on CK then reads

{{u ⊗ u}}=1⊗ u2 − u2 ⊗ 12

, {{v ⊗ v}}= v2 ⊗ 1− 1⊗ v2

2,

{{u ⊗ v}}=u ⊗ v − v ⊗ u − vu ⊗ 1− 1⊗ uv2

.

Proposition (S.A.’2016)Let K be an automorphism of CK defined on generators as

K :

{u → uvu−1,v → u−1 + v−1u−1.

Bracket {{, }} defined above is equivariant under the action of K

K({{a,b}}

)= {{K (a),K (b)}}

for all a,b ∈ CK .


Kontsevich system

Map K is a symmetry of the following system of noncommutative ODEdudt

= uv − uv−1 − v−1,

dvdt

= −vu + vu−1 + u−1.

Denote the induced H0-Poisson structure as

{, }K : A⊗ A→ A; ∀a,b ∈ A, {a,b}K = µ({{a,b}}K ).

Lemma (S.A.’2015)Noncommutative ODE defined above is a generalized Hamilton flow , namely

∀x ∈ A,dxdt

= {h, x}K , where h = u + v + u−1 + v−1 + u−1v−1.


Higher Hamilton flows

Proposition (S.A.’2015)There exists an infinite family of commuting flows, for all m, j ∈ N

ddtm

: A→ A,d

dtm(x) := {hm, x}K ;

[d

dtm,

ddtj

]= 0.


Example 2: Berenstein-Retakh NC 4-gon

Figure: Pair of hexagons glued together

It’s fundamental group is generated by oriented arcs xi±,j± subject to tworelations

R1 = x−10,3 x2,3x−1

2,0 x3,0x−13,2 x0,2 = 1,

R2 = x−10,2 x1,2x−1

1,0 x2,0x−12,1 x0,1 = 1.

(16)


Associated ordered cyclic graph

Total order of outgoing arcs at each vertex

0− : x0,3, x0,2, x0,11− : x1,0, x1,22− : x2,1, x2,0, x2,33− : x3,2, x3,0

0+ : x−13,0 , x

−12,0 , x

−11,0

1+ : x−10,1 , x

−12,1

2+ : x−11,2 , x

−10,2 , x

−13,2

3+ : x−12,3 , x

−10,3

(17)


Double brackets for NC 4-gon

{{xi,j ⊗ xk,l}}X =λ(i,j),(k,l)

2xk,l ⊗ xi,j (18a)

λ(i,j),(k,l) x0,3 x0,2 x0,1 x1,0 x1,2 x2,1 x2,0 x2,3 x3,2 x3,0

x0,3 1 1x0,2 −1 1x0,1 −1 −1x1,0 1x1,2 −1x2,1 1 1x2,0 −1 1x2,3 −1 −1x3,2 1x3,0 −1

(18b)


LemmaDouble Brackets (18) preserve the two-sided ideal generated by R1 and R2 inCX .

Checked explicitly on generators, for example

{{R2 ⊗ x0,3}}X =12

x0,3 ◦ R2 ⊗ Id0 −12

x0,3 ⊗ R2 = 0.

Now let F : CY → CX denote a functor corresponding to mutation of the 4-gon.On generators we have

F (yi,j ) =

xi,j , (3,1) 6= (i , j) 6= (1,3),

x0,3x−10,2 x1,2 + x2,3x−1

2,0 x1,0, (i , j) = (1,3),

x0,1x−10,2 x3,2 + x2,1x−1

2,0 x3,0, (i , j) = (3,1).

(19)

PropositionF defines a Homomorphism of a double quasi Poisson structures.


THE END

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Documents

Poisson Geometry of Noncommutative Cluster …One deﬁnes an algebra of polyvector ﬁelds as D = T C1(M)D 1. Puzzle: Der(A;A) is no longer an A-module for a noncommutative algebra