Cluster Structures on Drinfeld Doublesmath.berkeley.edu/~libland/gone-fishing-2014/gekhtman.pdf ·...

Cluster Structures on Drinfeld Doubles

M. Gekhtman ( joint with M. Shapiro and A. Vainshtein)

Gone Fishing 2014

M. Gekhtman ( joint with M. Shapiro and A. Vainshtein)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 1 / 25

Cluster Algebras

A seed (of geometric type) - a pair Σ = (x, B) :extended cluster x = (x1, . . . , xn︸︷︷︸

cluster

, xn+1, . . . , xn+m︸︷︷︸stable

extended exchange matrix B - an n × (n + m) integer matrix whosen × n principal part B is skew-symmetrizable.(Skew-symmetric case: B is an adjacency matrix of a quiver Q.)

The adjacent cluster in direction k ∈ [1, n]:

xk = (x \ {xk}) ∪ {x ′k},where the new cluster variable x ′k is given by the exchange relation

xkx′k =

∏1≤i≤n+m

xbkii +∏

1≤i≤n+mbki<0

x−bkii ;

B ′ is obtained from B by a matrix mutation in direction k:

b′ij =

−bij , if i = k or j = k;

bij +|bik |bkj + bik |bkj |

2, otherwise.

M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 2 / 25

Cluster Algebras

cluster

, xn+1, . . . , xn+m︸︷︷︸stable

xkx′k =

∏1≤i≤n+m

xbkii +∏

1≤i≤n+mbki<0

x−bkii ;

b′ij =

2, otherwise.

Cluster Algebras

A seed (of geometric type)

- a pair Σ = (x, B) :extended cluster x = (x1, . . . , xn︸︷︷︸

cluster

, xn+1, . . . , xn+m︸︷︷︸stable

xkx′k =

∏1≤i≤n+m

xbkii +∏

1≤i≤n+mbki<0

x−bkii ;

b′ij =

2, otherwise.

Cluster Algebras

cluster

, xn+1, . . . , xn+m︸︷︷︸stable

xkx′k =

∏1≤i≤n+m

xbkii +∏

1≤i≤n+mbki<0

x−bkii ;

b′ij =

2, otherwise.

Cluster Algebras

cluster

, xn+1, . . . , xn+m︸︷︷︸stable

xkx′k =

∏1≤i≤n+m

xbkii +∏

1≤i≤n+mbki<0

x−bkii ;

b′ij =

2, otherwise.

Cluster Algebras

cluster

, xn+1, . . . , xn+m︸︷︷︸stable

The adjacent cluster in direction k ∈ [1, n]

xkx′k =

∏1≤i≤n+m

xbkii +∏

1≤i≤n+mbki<0

x−bkii ;

b′ij =

2, otherwise.

Cluster Algebras

cluster

, xn+1, . . . , xn+m︸︷︷︸stable

xkx′k =

∏1≤i≤n+m

xbkii +∏

1≤i≤n+mbki<0

x−bkii ;

b′ij =

2, otherwise.

Cluster Algebras

cluster

, xn+1, . . . , xn+m︸︷︷︸stable

xkx′k =

∏1≤i≤n+m

xbkii +∏

1≤i≤n+mbki<0

x−bkii ;

b′ij =

2, otherwise.

Cluster Algebras

cluster

, xn+1, . . . , xn+m︸︷︷︸stable

xkx′k =

∏1≤i≤n+m

xbkii +∏

1≤i≤n+mbki<0

x−bkii ;

B ′ is obtained from B by a matrix mutation in direction k

b′ij =

2, otherwise.

Cluster Algebras

cluster

, xn+1, . . . , xn+m︸︷︷︸stable

xkx′k =

∏1≤i≤n+m

xbkii +∏

1≤i≤n+mbki<0

x−bkii ;

b′ij =

2, otherwise.

Σ′ = (x′, B ′) is called adjacent to Σ in direction k . Two seeds aremutation equivalent if they can be connected by a sequence ofpairwise adjacent seeds.

Cluster structure C(B) : The set of all seeds mutation equivalent toΣ.

Cluster algebra (of geometric type) A = A(B) is generated by allcluster variables in all seeds mutation equivalent to Σ .

Upper cluster algebra A = A(C) = A(B) is the intersection of therings of Laurent polynomials in cluster variables taken over all seedsin C(B).

Laurent phenomenon

A(C) ⊆ A(C)

Laurent phenomenon

A(C) ⊆ A(C)

Laurent phenomenon

A(C) ⊆ A(C)

Laurent phenomenon

A(C) ⊆ A(C)

Laurent phenomenon

A(C) ⊆ A(C)

Compatible Poisson Brackets

A Poisson bracket {·, ·} on FC = C(x1, . . . , xn+m) is compatible with thecluster algebra A if, for any extended cluster x = (x1, . . . , xn+m)

{xi , xj} = ωijxixj ,

where ωij ∈ Z are constants for all i , j ∈ [1, n + m].

Theorem (G.-S.-V.)

Assume that B is of full rank. Then there is a Poisson bracket compatiblewith A(B).

Theorem (G.-S.-V.)

Global Toric Action

Local toric action :

T Wd (xi ) = xi

r∏α=1

dwiαα , i ∈ [n + m], d = (d1, . . . , dr ) ∈ (C∗)r ,

where W = (wiα) is an integer (n + m)× r weight matrix of full rank.

Compatibility condition :

FC = C(x) −−−−→ FC = C(x′)

y yT W ′d

FC = C(x) −−−−→ FC = C(x′)

If local toric actions at all clusters are compatible, they define a globaltoric action Td on FC.

Global Toric Action

T Wd (xi ) = xi

r∏α=1

dwiαα , i ∈ [n + m], d = (d1, . . . , dr ) ∈ (C∗)r ,

FC = C(x) −−−−→ FC = C(x′)

y yT W ′d

FC = C(x) −−−−→ FC = C(x′)

Global Toric Action

T Wd (xi ) = xi

r∏α=1

dwiαα , i ∈ [n + m], d = (d1, . . . , dr ) ∈ (C∗)r ,

FC = C(x) −−−−→ FC = C(x′)

y yT W ′d

FC = C(x) −−−−→ FC = C(x′)

Key Observation

Let (V , {·, ·}) be a Poisson variety that

possesses a coordinate system x = (x1, . . . , xn+m) with Poissonrelations as above (log-canonical) for some ωij ∈ Z;

admits an action of (C∗)m that induces a local toric action of rank mon x.

Then there exists a unique skew-symmetric cluster structure C(B) with theinitial extended cluster x and stable variables xn+1, . . . , xn+m that iscompatible with {·, ·} and such that the local toric action above extendsto a global toric action.

Poisson-Lie Groups

(G, {·, ·}) is called a Poisson–Lie group if the multiplication map

G × G 3 (x , y) 7→ xy ∈ G

is Poisson.

We are interested in the case

G is a simple complex Lie group;

{·, ·} = {·, ·}r is associated with a classical R-matrix r - a solution ofthe CYBE:

{X⊗,X}r := [r ,X ⊗ X ]

Poisson-Lie Groups

G × G 3 (x , y) 7→ xy ∈ G

is Poisson.We are interested in the case

{X⊗,X}r := [r ,X ⊗ X ]

Poisson-Lie Groups

G × G 3 (x , y) 7→ xy ∈ G

is Poisson.We are interested in the case

{X⊗,X}r := [r ,X ⊗ X ]

Belavin-Drinfeld Classification

Up to an automorphism, every classical R-matrix r belongs to one ofdisjoint classes RT specified by the Belavin-Drinfeld data

T = (Γ1, Γ2, τ), (Γ1,2 ⊂ ∆, τ : Γ1 → Γ2),

where ∆ is the set of simple positive roots and τ is an isometry s.t.

∀α ∈ Γ1 ∃m ∈ N : τ j(α) ∈ Γ1 (j = 0, . . . ,m − 1), τm(α) /∈ Γ1 .

RT is linear space of dimension kT (kT−1)2 , where kT = dim hT ,

hT = {h ∈ h : α(h) = β(h) if β = τ j(α)},

T = (Γ1, Γ2, τ), (Γ1,2 ⊂ ∆, τ : Γ1 → Γ2),

∀α ∈ Γ1 ∃m ∈ N : τ j(α) ∈ Γ1 (j = 0, . . . ,m − 1), τm(α) /∈ Γ1 .

hT = {h ∈ h : α(h) = β(h) if β = τ j(α)},

T = (Γ1, Γ2, τ), (Γ1,2 ⊂ ∆, τ : Γ1 → Γ2),

∀α ∈ Γ1 ∃m ∈ N : τ j(α) ∈ Γ1 (j = 0, . . . ,m − 1), τm(α) /∈ Γ1 .

hT = {h ∈ h : α(h) = β(h) if β = τ j(α)},

Main Conjecture

Let G be a simple complex Lie group.For any Belavin-Drinfeld triple T = (Γ1, Γ2, τ) there exists a clusterstructure CT on G such that

the number of stable variables is 2 dim hT , and the correspondingextended exchange matrix has a full rank;

CT is regular, and the corresponding upper cluster algebra AC(CT ) isnaturally isomorphic to O(G);

the global toric action is generated by the action ofexp(hT )× exp(hT ) on G given by (H1,H2)(X ) = H1XH2;

for any r ∈ RT , {·, ·}r is compatible with CT ;

a Poisson–Lie bracket on G is compatible with CT only if it is a scalarmultiple {·, ·}r for some r ∈ RT .

Main Conjecture

Example I: Standard Case

Trivial Belavin-Drinfeld data : Γ1 = Γ2 = ∅l

Standard Poisson-Lie Structurel

Berenshtein-Fomin-Zelevinsky cluster structure on double Bruhat cells

Initial cluster (GLn case) : collection of all trailing dense minors

Example I: Standard Case

Trivial Belavin-Drinfeld data : Γ1 = Γ2 = ∅l

Standard Poisson-Lie Structurel

Berenshtein-Fomin-Zelevinsky cluster structure on double Bruhat cells

Initial cluster (GLn case) : collection of all trailing dense minors

Standard cluster structure in GL5: initial quiver

Example II: ”Maximal” Belavin-Drinfeld Data

Cremmer-Gervais Poisson Structure

G = SLnΓ1 = {α2, . . . , αn−1}, Γ2 = {α1, . . . , αn−2}

γ(αi ) = αi−1 for i = 2, . . . , n − 1.

Theorem

There exists a cluster structure CCG on SLn/GLn/MatN compatible withthe Cremmer–Gervais Poisson–Lie structure and satisfying all conditions ofthe Main Conjecture.

G = SLnΓ1 = {α2, . . . , αn−1}, Γ2 = {α1, . . . , αn−2}

γ(αi ) = αi−1 for i = 2, . . . , n − 1.

Theorem

G = SLnΓ1 = {α2, . . . , αn−1}, Γ2 = {α1, . . . , αn−2}

γ(αi ) = αi−1 for i = 2, . . . , n − 1.

Theorem

Table: Cremmer-Gervais vs. Standard Poisson-Lie bracket

Standard Cremmer-Gervais

{x11, x55} 2x15x51 x15x51 + x21x45 + x25x41 + x21x45 + x31x35{x12, x52} x12x52

15x12x52 + 2x22x42 + x232 − x11x53 + x13x51

{x15, x51} x12x52 −35x15x51 + x21x45 + x25x41 + x31x35

Initial Cluster

For X ,Y ∈ Matn, let

X =[X[2,n] 0

], Y =

[0 Y[1,n−1]

Put k = bn+12 c, N = k(n − 1) and define a k(n − 1)× (k + 1)(n + 1)

matrix

U(X ,Y ) =

Y X 0 · · · 00 Y X 0 · · ·

0. . .

. . .. . . 0

0 · · · 0 Y X

.Define

θi (X ) = detX[n−i+1,n][n−i+1,n] , i ∈ [n − 1];

ϕp(X ,Y ) = detU(X ,Y )[k(n+1)−p+1,k(n+1)][N−p+1,N] , p ∈ [N];

ψq(X ,Y ) = detU(X ,Y )[k(n+1)−q+2,k(n+1)+1][N−q+1,N] , q ∈ [M].

In the last family, M = N/M = N − n + 1 if n is even/odd.

Initial Cluster

X =[X[2,n] 0

], Y =

[0 Y[1,n−1]

matrix

U(X ,Y ) =

Y X 0 · · · 00 Y X 0 · · ·

0. . .

. . .. . . 0

0 · · · 0 Y X

.Define

Initial Cluster

X =[X[2,n] 0

], Y =

[0 Y[1,n−1]

matrix

U(X ,Y ) =

Y X 0 · · · 00 Y X 0 · · ·

0. . .

. . .. . . 0

0 · · · 0 Y X

Define

Initial Cluster

X =[X[2,n] 0

], Y =

[0 Y[1,n−1]

matrix

U(X ,Y ) =

Y X 0 · · · 00 Y X 0 · · ·

0. . .

. . .. . . 0

0 · · · 0 Y X

.Define

In the last family, M = N/M = N − n + 1 if n is even/odd.M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 14 / 25

Figure: Translation invariance properties of U(X ,X )

Theorem

The functions θi (X ), φp(X ,X ), ψq(X ,X ) form a log-canonical family withrespect to the Cremmer–Gervais bracket.

Intuition behind a construction of the initial cluster as well as the methodof the proof come from considering the Poisson-Lie (Drinfeld) double ofSLN associated with the Cremmer-Gervais structure.

Theorem

The functions θi (X ), φp(X ,X ), ψq(X ,X ) form a log-canonical family withrespect to the Cremmer–Gervais bracket.

Intuition behind a construction of the initial cluster as well as the methodof the proof come from considering the Poisson-Lie (Drinfeld) double ofSLN associated with the Cremmer-Gervais structure.

Initial Quiver

Figure: Quiver QCG (5)

Theorem

The cluster structure CCG is regular.

The proof relies on Dodgson-type identities applied to submatrices ofU(X ,Y ) while taking into account its shift-invariance properties.

Theorem

The cluster structure CCG is regular.

The proof relies on Dodgson-type identities applied to submatrices ofU(X ,Y ) while taking into account its shift-invariance properties.

Theorem

O(Matn) ⊂ A(CCG )

The proof relies on induction on n.

Strategy

Two distinguished sequences of cluster transformations:

S (# of mutations quadratic in n) - followed by freezing some of thecluster variables and localization at a single cluster variable

ϕn−1(X ) = detX[2,n][1,n−1] - realizes a map

ζ: Matn \{X : ϕn−1(X ) = 0} → Matn−1

that “respects” the Cremmer–Gervais cluster structure.

T (# of mutations cubic in n) - realizes the anti-Poisson involutionX 7→W0XW0 ( W0 - the longest permutation)

Theorem

O(Matn) ⊂ A(CCG )

Strategy

ζ: Matn \{X : ϕn−1(X ) = 0} → Matn−1

Theorem

O(Matn) ⊂ A(CCG )

Strategy

ζ: Matn \{X : ϕn−1(X ) = 0} → Matn−1

Theorem

O(Matn) ⊂ A(CCG )

Strategy

ζ: Matn \{X : ϕn−1(X ) = 0} → Matn−1

Theorem

O(Matn) ⊂ A(CCG )

Strategy

ζ: Matn \{X : ϕn−1(X ) = 0} → Matn−1

Further Results and Work in Progress

Theorem

TotPosCG (n) ( TotPos(n) .

Theorem

The cluster algebra ACG (3) is a proper subalgebra of the upper clusteralgebra ACG (3).

Idea of the proof: show that x12 can not belong to a log-canonicalcoordinate chart w.r.t. the Cremmer-Gervais Poisson structure.

Conjecture

The cluster algebra ACG (n) is a proper subalgebra of the upper clusteralgebra ACG (n).

Theorem

Conjecture

Theorem

Conjecture

Theorem

Conjecture

For any Belavin-Drinfeld data, there exists a compatible generalized clusterstructure on the corresponding Drinfeld double and the dual Poisson-Liegroup.

Proved for both the standard and Cremmer-Gervais cases in GLn.

General GLn case: proof in progress.

Conjecture

For any Belavin-Drinfeld data, there exists a compatible generalized clusterstructure on the corresponding Drinfeld double and the dual Poisson-Liegroup.

Proved for both the standard and Cremmer-Gervais cases in GLn.

General GLn case: proof in progress.

Example in GL8:

Initial quiver for the standard double of GL4:

References

1 Cluster structures on simple complex Lie groups and Belavin-Drinfeldclassification, Mosc. Math. J. 12 (2012), no. 2, 293–312.

2 Exotic cluster structures on SLn: the Cremmer-Gervais case,arXiv:1307.1020.

3 Cremmer-Gervais cluster structure on on SLn, PNAS 2014.

Thank you!

Cluster Structures on Drinfeld Doublesmath.berkeley.edu/~libland/gone-fishing-2014/gekhtman.pdf ·...

Documents

Transcript of Cluster Structures on Drinfeld Doublesmath.berkeley.edu/~libland/gone-fishing-2014/gekhtman.pdf ·...