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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
bki>0
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
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
bki>0
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
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
bki>0
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
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
bki>0
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
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
bki>0
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
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
bki>0
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
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
bki>0
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
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
bki>0
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
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
bki>0
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
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
bki>0
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
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)
M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 3 / 25
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)
M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 3 / 25
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)
M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 3 / 25
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)
M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 3 / 25
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)
M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 3 / 25
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).
M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 4 / 25
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).
M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 4 / 25
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).
M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 4 / 25
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′)
T Wd
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.
M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 5 / 25
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′)
T Wd
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.
M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 5 / 25
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′)
T Wd
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.
M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 5 / 25
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.
M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 6 / 25
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 ]
M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 7 / 25
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 ]
M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 7 / 25
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 ]
M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 7 / 25
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)},
M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 8 / 25
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)},
M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 8 / 25
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)},
M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 8 / 25
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 .
M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 9 / 25
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 .
M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 9 / 25
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 .
M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 9 / 25
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 .
M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 9 / 25
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 .
M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 9 / 25
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 .
M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 9 / 25
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
M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 10 / 25
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
M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 10 / 25
Standard cluster structure in GL5: initial quiver
M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 11 / 25
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.
M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 12 / 25
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.
M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 12 / 25
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.
M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 12 / 25
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
M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 13 / 25
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.
M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 14 / 25
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.
M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 14 / 25
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.
M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 14 / 25
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.M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 14 / 25
τ1
τ2
τ3
Figure: Translation invariance properties of U(X ,X )
M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 15 / 25
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.
M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 16 / 25
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.
M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 16 / 25
Initial Quiver
θ5
θ3
ϕ8
ψ7
ψ6
ϕ4
ϕ3
11
ψ4
ϕ10
12ϕ
ϕ6
ψ1
ϕ5 5
ψ
ϕ2
ψ2
ϕ1
θ1
θ2
ϕ9
ψ8
ϕ7
ψ3
ϕ
θ4
Figure: Quiver QCG (5)
M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 17 / 25
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.
M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 18 / 25
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.
M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 18 / 25
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)
M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 19 / 25
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)
M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 19 / 25
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)
M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 19 / 25
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)
M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 19 / 25
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)
M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 19 / 25
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).
M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 20 / 25
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).
M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 20 / 25
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).
M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 20 / 25
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).
M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 20 / 25
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.
M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 21 / 25
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.
M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 21 / 25
Example in GL8:
M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 22 / 25
Initial quiver for the standard double of GL4:
M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 23 / 25
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.
M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 24 / 25
Thank you!
M. Gekhtman ( joint with M. Shapiro and A. Vainshtein) (Notre Dame)Cluster Structures on Drinfeld Doubles Gone Fishing 2014 25 / 25
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