Noncommutative Poisson Geometry and Cluster Integrable Systems S. - - PowerPoint PPT Presentation

Noncommutative Poisson Geometry and Cluster Integrable Systems

• S. Arthamonov

Rutgers, The State University of New Jersey May 7, 2018 Cluster Algebras and Mathematical Physics

• S. Arthamonov

NC Poisson Geometry and Cluster Algebras CAMP-2018, East Lansing MI 1 / 26

Cluster algebras associated to triangulated surfaces

Figure: Flip of an ideal triangulation.

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

Mathematics, 328:1010–1087, 2018.

• S. Arthamonov

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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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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

NC Poisson Geometry and Cluster Algebras CAMP-2018, East Lansing MI 4 / 26

Ideal triangulations and bipartite graphs

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

Definition

A conjugate surface ˆ SΓ associated to the ribbon graph Γ is a surface corresponding to the ribbon graph with reversed cyclic order of edges at each vertex. Both ˆ SΓ and SΓ have the same fundamental group as the underlying graph π1( ˆ SΓ) = π1(SΓ) = π1(Γ). (1) The identification (1) allows one to introduce two different Poisson structures

• n the character variety of π1(Γ).
• S. Arthamonov

NC Poisson Geometry and Cluster Algebras CAMP-2018, East Lansing MI 5 / 26

Poisson bracket on graph connections

Each 1-dimensional representation ϕ ∈ Hom(π1(SΓ), C×) is determined by x1 = ϕ(M1), . . . , xn = ϕ(Mn). We can equip C[x1, . . . , xn] with a Poisson bracket as follows {, } :

• C[x1, . . . , xn]

⊗2 → C[x1, . . . , xn], {xi, xj} =

• p

ǫi,j(p)xixj, ǫij(p) =                    +1 Mj Mi p −1 Mi Mj p

• S. Arthamonov

NC Poisson Geometry and Cluster Algebras CAMP-2018, East Lansing MI 6 / 26

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

NC Poisson Geometry and Cluster Algebras CAMP-2018, East Lansing MI 7 / 26

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

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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

NC Poisson Geometry and Cluster Algebras CAMP-2018, East Lansing MI 9 / 26

Vector fields on a linear category

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

NC Poisson Geometry and Cluster Algebras CAMP-2018, East Lansing MI 10 / 26

Modules over a linear category

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

NC Poisson Geometry and Cluster Algebras CAMP-2018, East Lansing MI 11 / 26

Category of 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

NC Poisson Geometry and Cluster Algebras CAMP-2018, East Lansing MI 12 / 26

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

NC Poisson Geometry and Cluster Algebras CAMP-2018, East Lansing MI 13 / 26

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

NC Poisson Geometry and Cluster Algebras CAMP-2018, East Lansing MI 14 / 26

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

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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.

• V. Fock and A. Rosly. Poisson structure on moduli of flat connections on Riemann

surfaces and r-matrix In Moscow seminar in math. phys., pp. 67-–86. AMS, 1999.

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

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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

NC Poisson Geometry and Cluster Algebras CAMP-2018, East Lansing MI 18 / 26

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) Representations of A then form an affine scheme V with a coordinate ring C[V] := C

• x(i)

j,k

• /ϕ(R). Denote as CV — the corresponding sheaf of rational

functions.

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

• S. Arthamonov

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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 CV ⊗ CV 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.
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Induced Poisson Bracket

Proposition

The following restriction { , }V : CGLN(C)

V

⊗ CV → CV (10) satisfies the left Loday-Jacobi identity: for all f, g ∈ CGLN(C)

V

and h ∈ CV : {f, {g, h}V}V − {g, {f, h}V}V = {{f, g}V, h}V. (11) For all f, g ∈ CGLN(C)

V

we have {f, g}V ∈ CGLN(C)

V

and {f, g}V = −{g, f}V.

Proposition

The following restriction of {, }V {, }inv : CGLN(C)

V

⊗ CGLN(C)

V

→ CGLN(C)

V

(13) is a Poisson bracket.

• S. Arthamonov

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Simpelst example: Kronecker quiver

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

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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

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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

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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

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THE END

Thank you for your attention!

• S. Arthamonov

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