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Poisson Geometry of Noncommutative Cluster Algebras

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 and

Weil-Petersson forms. Duke Mathematical Journal, 127(2), 291-311, 2005.

• V. Fock, A. Goncharov, A. Moduli spaces of local systems and higher Teichm¨

uller

• theory. Publications Math´

ematiques de l’IH ´ ES, 103, 1-211, 2006.

• S. Fomin, M. Shapiro, and D. Thurston. Cluster algebras and triangulated
• surfaces. Part I: Cluster complexes. Acta Mathematica, 201(1):83–146, 2007.
• A. Goncharov, and R. Kenyon. Dimers and cluster integrable systems. Annales

scientifiques de l’ ´ Ecole Normale Sup´

• erieure. Vol. 46. No. 5., 2013.
• S. Arthamonov

Poisson Geometry of NC Cluster Algebras AMS Sectional Meeting, UC Riverside 2 / 32

Ribbon Graphs

Definition

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

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

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

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

• S. Arthamonov

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Ideal triangulations and bipartite graphs

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

Definition

A conjugate surface ˆ SΓ associated to the bipartite ribbon graph Γ is a surface corresponding to the ribbon graph with reversed cyclic order of edges at each black vertex. Surface ˆ SΓ has the same fundamental group as the underlying graph π1( ˆ SΓ) = π1(Γ). (1)

• S. Arthamonov

Poisson Geometry of NC Cluster Algebras AMS Sectional Meeting, UC Riverside 4 / 32

Graph connections and edge weights

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

Definition

Graph connection is an assignment of a parallel transport ge ∈ G to each

• riented edge e ∈ E(Γ).

The gauge group GV(Γ) = {V(Γ) → G, v → hv} acts on a graph connections ge → ht(e)geh−1

s(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).

• S. Arthamonov

Poisson Geometry of NC Cluster Algebras AMS Sectional Meeting, UC Riverside 5 / 32

Cluster algebras and Poisson Geometry

Let k be a ground field of characteristic zero.

Definition

Let 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 bracket compatible with mutations.

• M. Gekhtman, M. Shapiro, and A. Vainshtein. Cluster algebras and Poisson
• geometry. Moscow Mathematical Journal, 3(3):899–934, 2003.
• S. Arthamonov

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Poisson bracket on graph connections

Each 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

ǫi,j(p)yiyj, ǫij(p) =                  +1 Mj Mi p −1 Mi Mj p M1 M2 M3

• S. Arthamonov

Poisson Geometry of NC Cluster Algebras AMS Sectional Meeting, UC Riverside 7 / 32

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−1

0 ,

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

0 )−1,

i = 2, 4.

• S. Arthamonov

Poisson Geometry of NC Cluster Algebras AMS Sectional Meeting, UC Riverside 8 / 32

Noncommutative cluster algebras

Cluster algebras with noncommutative edge weights

• A. Berenstein, V. Retakh Noncommutative marked surfaces. Advances in

Mathematics, 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 every noncommutative 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 a cluster 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 of

Pure and Applied Algebra, 125(1–3):155 – 190, 1998.

• S. Arthamonov

Poisson Geometry of NC Cluster Algebras AMS Sectional Meeting, UC Riverside 9 / 32

Algebra of polyvector fields

A vector field d on M can be viewed as the derivation of C∞(M), the algebra

• f smooth functions on M

d : C∞(M) → C∞(M), d(fg) = fd(g) + d(f)g, for all f, g ∈ C∞(M).

Lemma

The 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.

• S. Arthamonov

Poisson Geometry of NC Cluster Algebras AMS Sectional Meeting, UC Riverside 10 / 32

Double Geometry

• W. Crawley-Boevey, P

. Etingof, and V. Ginzburg. Noncommutative geometry and quiver algebras. Advances in Mathematics, 209(1):274 – 336, 2007

• M. Van den Bergh. Double Poisson algebras. Trans. Amer. Math. Soc.,

360:5711–5769, 2008.

Definition

Let A be an associative algebra. We say that map δ is a noncommutative vector field if δ : A → A ⊗ A, δ(ab) = (a ⊗ 1)δ(b) + δ(a)(1 ⊗ b) for all a, b ∈ A.

Lemma

Noncommutative vector fields DA = Der(A, A ⊗ A) form an A-bimodule. One defines a noncommutative algebra of polyvector fields as TADA

• S. Arthamonov

Poisson Geometry of NC Cluster Algebras AMS Sectional Meeting, UC Riverside 11 / 32

Double derivations of a ring with many objects

Definition

Let 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 morphisms

• n V and W.

In what follows we denote the space of (V, W)-vector fields as D1

V,W. Let

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

Lemma

D1 is a covariant functor on C × Cop.

• S. Arthamonov

Poisson Geometry of NC Cluster Algebras AMS Sectional Meeting, UC Riverside 12 / 32

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, the tensor product R ⊗C L is defined as R ⊗C L =  

V∈Obj C

RV ⊗ LV  

• ρ◦f⊗λ∼ρ⊗f◦λ

.

Definition (Trace)

Let M be a bifunctor on C, the trace over C is defined as trC : M → M♮ :=  

X∈Obj C

MX,X  

• f◦m∼m◦f

.

• S. Arthamonov

Poisson Geometry of NC Cluster Algebras AMS Sectional Meeting, UC Riverside 13 / 32

Polyvector fields

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

V,W =

• U1,...,Uk−1∈Obj C

D1

V,U1 ⊗C

. . . ⊗C D1

Uk−1,W,

where for k = 0 we assume that D0

V,W = hom(W, V).

Corollary

Dk is a covariant functor on C × Cop. D•

V,W = ∞

• k=0

Dk

V,W

We define the category V of polyvector fields on C as Obj V = Obj C, homV(W, V) = D•

V,W.

• S. Arthamonov

Poisson Geometry of NC Cluster Algebras AMS Sectional Meeting, UC Riverside 14 / 32

Traces of polyvector fields and polyderivations

Let δ1, . . . , δk be a chain of composable vector fields. The trace trC(δ1 ⋆ · · · ⋆ δk) is equivalent to the following map trC(δ1 ⋆ · · · ⋆ δk) : (Mor C)⊗k → (Mor C)⊗k, f1 ⊗ · · · ⊗ fk → (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))

• S. Arthamonov

Poisson Geometry of NC Cluster Algebras AMS Sectional Meeting, UC Riverside 15 / 32

Double Quasi Poisson Bracket on a category

Definition

A 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)
• p

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.

• S. Arthamonov

Poisson Geometry of NC Cluster Algebras AMS Sectional Meeting, UC Riverside 16 / 32

Category associated to a ribbon graph

C0 = k π1( ˆ SΓ, V1, . . . , Vn). The objects Obj C0 = {Vi} correspond to marked points. x3 x1 x2

(a) Disk corresponding to white vertex

f1 f2 f −1

1

• f −1

2

(b) Disc corresponding to black vertex

Figure: Building blocks for bipartite graph with trivalent black vertices

• S. Arthamonov

Poisson Geometry of NC Cluster Algebras AMS Sectional Meeting, UC Riverside 17 / 32

Double Quasi Poisson Bivector

For each object V ∈ C0 consider a noncommutative bivector PV = 1 2

• 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 = j.

Lemma

The following map is a double Quasi Poisson Bracket on C0 { {, } }=

• V∈Obj C0

trC0PV.

• G. Massuyeau and V. Turaev. Quasi-Poisson structures on representation spaces
• f surfaces. IMRN, 2014(1):1–64, 2012.
• S. Arthamonov

Poisson Geometry of NC Cluster Algebras AMS Sectional Meeting, UC Riverside 18 / 32

Noncommutative rectangle move

Let Csub

1

⊂ C1 be a subcategory generated by Y ±1

1

, Y ±1

2

, Y ±1

3

, Y ±1

4

. Similarly we define a subcategory Csub

2

⊂ 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

• S. Arthamonov

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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 −1

1

, 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.

• S. Arthamonov

Poisson Geometry of NC Cluster Algebras AMS Sectional Meeting, UC Riverside 20 / 32

Representation Scheme

Following general philosophy by M. Kontsevich any algebraic property that makes 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 Gelfand Mathematical Seminars, 1990–1992, pages 173–187. Birkhauser Boston, 1993.

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

• S. Arthamonov

Poisson Geometry of NC Cluster Algebras AMS Sectional Meeting, UC Riverside 21 / 32

Induced Brackets on Representation Scheme

Let { {, } } be a double Quasi Poisson bracket. Define induced bracket {, }V on generators 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-Poisson
• manifolds. Canad. J. Math., (54):3–29, 2002.
• S. Arthamonov

Poisson Geometry of NC Cluster Algebras AMS Sectional Meeting, UC Riverside 22 / 32

Induced Poisson Bracket

Proposition

The 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.

Proposition

The following restriction of {, }V {, }inv : C[V]GLN ⊗ C[V]GLN → C[V]GLN (13) is a Poisson bracket.

• S. Arthamonov

Poisson Geometry of NC Cluster Algebras AMS Sectional Meeting, UC Riverside 23 / 32

Example 1: Kontsevich system

a b c

(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 = ku±1, v±1 becomes a group algebra of a π1(T\D).

• S. Arthamonov

Poisson Geometry of NC Cluster Algebras AMS Sectional Meeting, UC Riverside 24 / 32

Bracket on a torus and Kontsevich map

Bracket on CK then reads { {u ⊗ u} }=1 ⊗ u2 − u2 ⊗ 1 2 , { {v ⊗ v} }= v2 ⊗ 1 − 1 ⊗ v2 2 , { {u ⊗ v} }=u ⊗ v − v ⊗ u − vu ⊗ 1 − 1 ⊗ uv 2 .

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.

• S. Arthamonov

Poisson Geometry of NC Cluster Algebras AMS Sectional Meeting, UC Riverside 25 / 32

Kontsevich system

Map K is a symmetry of the following system of noncommutative ODE        du dt = uv − uv−1 − v−1, dv dt = −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, dx dt = {h, x}K, where h = u + v + u−1 + v−1 + u−1v−1.

• S. Arthamonov

Poisson Geometry of NC Cluster Algebras AMS Sectional Meeting, UC Riverside 26 / 32

Higher Hamilton flows

Proposition (S.A.’2015)

There exists an infinite family of commuting flows, for all m, j ∈ N d dtm : A → A, d dtm (x) := {hm, x}K; d dtm , d dtj

• = 0.
• S. Arthamonov

Poisson Geometry of NC Cluster Algebras AMS Sectional Meeting, UC Riverside 27 / 32

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 two relations R1 = x−1

0,3x2,3x−1 2,0x3,0x−1 3,2x0,2 = 1,

R2 = x−1

0,2x1,2x−1 1,0x2,0x−1 2,1x0,1 = 1.

(16)

• S. Arthamonov

Poisson Geometry of NC Cluster Algebras AMS Sectional Meeting, UC Riverside 28 / 32

Associated ordered cyclic graph

Total order of outgoing arcs at each vertex 0− : x0,3, x0,2, x0,1 1− : x1,0, x1,2 2− : x2,1, x2,0, x2,3 3− : x3,2, x3,0 0+ : x−1

3,0, x−1 2,0, x−1 1,0

1+ : x−1

0,1, x−1 2,1

2+ : x−1

1,2, x−1 0,2, x−1 3,2

3+ : x−1

2,3, x−1 0,3

(17)

• S. Arthamonov

Poisson Geometry of NC Cluster Algebras AMS Sectional Meeting, UC Riverside 29 / 32

Double brackets for NC 4-gon

{ {xi,j ⊗ xk,l} }X= λ(i,j),(k,l) 2 xk,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 1 x0,2 −1 1 x0,1 −1 −1 x1,0 1 x1,2 −1 x2,1 1 1 x2,0 −1 1 x2,3 −1 −1 x3,2 1 x3,0 −1 (18b)

• S. Arthamonov

Poisson Geometry of NC Cluster Algebras AMS Sectional Meeting, UC Riverside 30 / 32

Lemma

Double Brackets (18) preserve the two-sided ideal generated by R1 and R2 in CX. Checked explicitly on generators, for example { {R2 ⊗ x0,3} }X= 1 2x0,3 ◦ R2 ⊗ Id0 − 1 2x0,3 ⊗ 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) = (i, j) = (1, 3), x0,3x−1

0,2x1,2 + x2,3x−1 2,0x1,0,

(i, j) = (1, 3), x0,1x−1

0,2x3,2 + x2,1x−1 2,0x3,0,

(i, j) = (3, 1). (19)

Proposition

F defines a Homomorphism of a double quasi Poisson structures.

• S. Arthamonov

Poisson Geometry of NC Cluster Algebras AMS Sectional Meeting, UC Riverside 31 / 32

THE END

Thank you for your attention!

• S. Arthamonov

Poisson Geometry of NC Cluster Algebras AMS Sectional Meeting, UC Riverside 32 / 32