Polynomial Poisson Algebras, Double Bruhat
Cells, and Symplectic Groupoids
Jianghua Lu
University of Hong Kong
QQQ2016, Kristineberg, Sweden
July 11 - 15, 2016
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Outline of talk
T -Poisson CGLs (Cauchon-Goodearl-Letzler), where T is a
torus, are a class of very special Poisson structures on
polynomial algebras
A = k[x1 , x2 , . . . , xn ].
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Outline of talk
T -Poisson CGLs (Cauchon-Goodearl-Letzler), where T is a
torus, are a class of very special Poisson structures on
polynomial algebras
A = k[x1 , x2 , . . . , xn ].
They were introduced by Goodearl, Launois, Oh, Yakimov ....
starting around 2007.
2 / 30
Outline of talk
T -Poisson CGLs (Cauchon-Goodearl-Letzler), where T is a
torus, are a class of very special Poisson structures on
polynomial algebras
A = k[x1 , x2 , . . . , xn ].
They were introduced by Goodearl, Launois, Oh, Yakimov ....
starting around 2007.
The following aspects of Poisson CGLs have been investigated:
2 / 30
Outline of talk
T -Poisson CGLs (Cauchon-Goodearl-Letzler), where T is a
torus, are a class of very special Poisson structures on
polynomial algebras
A = k[x1 , x2 , . . . , xn ].
They were introduced by Goodearl, Launois, Oh, Yakimov ....
starting around 2007.
The following aspects of Poisson CGLs have been investigated:
1
T -prime Poisson ideals (Goodearl, Launois, Lenagan, Oh, and
others);
2 / 30
Outline of talk
T -Poisson CGLs (Cauchon-Goodearl-Letzler), where T is a
torus, are a class of very special Poisson structures on
polynomial algebras
A = k[x1 , x2 , . . . , xn ].
They were introduced by Goodearl, Launois, Oh, Yakimov ....
starting around 2007.
The following aspects of Poisson CGLs have been investigated:
1
T -prime Poisson ideals (Goodearl, Launois, Lenagan, Oh, and
others);
2
cluster algebras and quantum cluster algebras
(Goodearl-Yaminov);
2 / 30
Outline of talk
T -Poisson CGLs (Cauchon-Goodearl-Letzler), where T is a
torus, are a class of very special Poisson structures on
polynomial algebras
A = k[x1 , x2 , . . . , xn ].
They were introduced by Goodearl, Launois, Oh, Yakimov ....
starting around 2007.
The following aspects of Poisson CGLs have been investigated:
1
T -prime Poisson ideals (Goodearl, Launois, Lenagan, Oh, and
others);
2
cluster algebras and quantum cluster algebras
(Goodearl-Yaminov);
3
examples, including semi-classical limits of quantum matrices,
quantum symmetric, quantum antisymmetric matrices,
quantum Euclidean spaces.
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Outline of talk
Today we will talk about
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Outline of talk
Today we will talk about
A geometrical setting where they arise. Namely, projective
Poisson manifolds paved by Poisson CGLs;
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Outline of talk
Today we will talk about
A geometrical setting where they arise. Namely, projective
Poisson manifolds paved by Poisson CGLs;
Double Bruhat cells (examples of Poisson CGLs) as Poisson
groupoids over Bruhat cells (also examples of Poisson CGLs).
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Section 1: Review on Poisson algebras and Poisson manifolds
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Basics of Poisson algebras
A Poisson algebra over a field k is a commutative k-algebra A
together with a k-bilinear skew-symmetric map
{, } : A × A → A, called the Poisson bracket, such that
Leibniz rule :
Jacobi identity :
{a, bc} = b{a, c} + c{a, b},
{a, {b, c}} + {b, {c, a}} + {c, {a, b}} = 0.
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Basics of Poisson algebras
A Poisson algebra over a field k is a commutative k-algebra A
together with a k-bilinear skew-symmetric map
{, } : A × A → A, called the Poisson bracket, such that
Leibniz rule :
Jacobi identity :
{a, bc} = b{a, c} + c{a, b},
{a, {b, c}} + {b, {c, a}} + {c, {a, b}} = 0.
A Poisson variety is a variety X with a Poisson algebra
structure on its structure sheaf.
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Basics of Poisson algebras
A Poisson algebra over a field k is a commutative k-algebra A
together with a k-bilinear skew-symmetric map
{, } : A × A → A, called the Poisson bracket, such that
Leibniz rule :
Jacobi identity :
{a, bc} = b{a, c} + c{a, b},
{a, {b, c}} + {b, {c, a}} + {c, {a, b}} = 0.
A Poisson variety is a variety X with a Poisson algebra
structure on its structure sheaf.
Let P be a “space” and let A = Fun(P). If there exists an
associative product ∗ on A[h] such that
a ∗ b = ab + hF1 (a, b) + h2 F2 (a, b) + · · · ,
a, b ∈ A,
then
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Basics of Poisson algebras
A Poisson algebra over a field k is a commutative k-algebra A
together with a k-bilinear skew-symmetric map
{, } : A × A → A, called the Poisson bracket, such that
Leibniz rule :
Jacobi identity :
{a, bc} = b{a, c} + c{a, b},
{a, {b, c}} + {b, {c, a}} + {c, {a, b}} = 0.
A Poisson variety is a variety X with a Poisson algebra
structure on its structure sheaf.
Let P be a “space” and let A = Fun(P). If there exists an
associative product ∗ on A[h] such that
a ∗ b = ab + hF1 (a, b) + h2 F2 (a, b) + · · · ,
a, b ∈ A,
then
{a, b} := F1 (a, b) − F1 (b, a),
a, b ∈ A,
is a Poisson bracket on A,
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Basics of Poisson algebras
A Poisson algebra over a field k is a commutative k-algebra A
together with a k-bilinear skew-symmetric map
{, } : A × A → A, called the Poisson bracket, such that
Leibniz rule :
Jacobi identity :
{a, bc} = b{a, c} + c{a, b},
{a, {b, c}} + {b, {c, a}} + {c, {a, b}} = 0.
A Poisson variety is a variety X with a Poisson algebra
structure on its structure sheaf.
Let P be a “space” and let A = Fun(P). If there exists an
associative product ∗ on A[h] such that
a ∗ b = ab + hF1 (a, b) + h2 F2 (a, b) + · · · ,
a, b ∈ A,
then
{a, b} := F1 (a, b) − F1 (b, a),
a, b ∈ A,
is a Poisson bracket on A, the semi-classical limit of (A[h], ∗).
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Basics of Poisson algebras
A Poisson algebra over a field k is a commutative k-algebra A
together with a k-bilinear skew-symmetric map
{, } : A × A → A, called the Poisson bracket, such that
Leibniz rule :
Jacobi identity :
{a, bc} = b{a, c} + c{a, b},
{a, {b, c}} + {b, {c, a}} + {c, {a, b}} = 0.
A Poisson variety is a variety X with a Poisson algebra
structure on its structure sheaf.
Let P be a “space” and let A = Fun(P). If there exists an
associative product ∗ on A[h] such that
a ∗ b = ab + hF1 (a, b) + h2 F2 (a, b) + · · · ,
a, b ∈ A,
then
{a, b} := F1 (a, b) − F1 (b, a),
a, b ∈ A,
is a Poisson bracket on A, the semi-classical limit of (A[h], ∗).
Today’s examples all come from quantum groups.
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Remarks
Leibniz’s rule =⇒ {, } is determined by the
n
2
functions
πij = {xi , xj } ∈ A for 1 ≤ i < j ≤ n, namely,
X
∂f ∂g
∂g ∂f
{f , g } =
{xi , xj }
−
.
∂xi ∂xj
∂xi ∂xj
1≤i<j≤n
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Remarks
Leibniz’s rule =⇒ {, } is determined by the
n
2
functions
πij = {xi , xj } ∈ A for 1 ≤ i < j ≤ n, namely,
X
∂f ∂g
∂g ∂f
{f , g } =
{xi , xj }
−
.
∂xi ∂xj
∂xi ∂xj
1≤i<j≤n
Jacobi’s identity =⇒ the πij ’s must satisfy the
n
X
m=1
πm,k
n
3
equations
∂πk,i
∂πj,k
∂πi,j
+ πm,j
+ πm,i
= 0, 1 ≤ i < j < k ≤ n,
∂xm
∂xm
∂xm
an over-determined system, not easy to make up artificially.
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Examples
Example 0. {xi , xj } = cij ∈ k, for any 1 ≤ i < j ≤ n.
Example 1. On A = k[x1 , x2 , x3 ],
{x1 , x2 } = x3 , {x2 , x3 } = x1 , {x3 , x1 } = x2 .
If cijk are the structure constants for an n-dimensional Lie algebra g
over k, one has a linear Poisson bracket on A = k[x1 , x2 , . . . , xn ] by
{xi , xj } =
n
X
cijk xk , 1 ≤ i, j ≤ n.
k=1
Example 2. For an n × n skew-symmetric matrix C = (cij ), one has
the log-canonical Poisson bracket on A = k[x1 , x2 , . . . , xn ]
{xi , xj } = cij xi xj
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Some geometric concepts
A complex Poisson manifold is a complex manifold P with a
holomorphic section π of ∧2 T (1,0) P, such that {f , g } = π(df , dg )
is a Poisson bracket on local holomorphic functions.
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Some geometric concepts
A complex Poisson manifold is a complex manifold P with a
holomorphic section π of ∧2 T (1,0) P, such that {f , g } = π(df , dg )
is a Poisson bracket on local holomorphic functions.
Symplectic leaves of a complex Poisson manifold (P, π) are the
integral submanifolds of the bundle map
π # : (T (1,0) P)∗ −→ T (1,0) P.
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Some geometric concepts
A complex Poisson manifold is a complex manifold P with a
holomorphic section π of ∧2 T (1,0) P, such that {f , g } = π(df , dg )
is a Poisson bracket on local holomorphic functions.
Symplectic leaves of a complex Poisson manifold (P, π) are the
integral submanifolds of the bundle map
π # : (T (1,0) P)∗ −→ T (1,0) P.
A T -Poisson manifold, where T is a complex torus, is a Poisson
manifold (P, π) with a T -action preserving π.
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Some geometric concepts
A complex Poisson manifold is a complex manifold P with a
holomorphic section π of ∧2 T (1,0) P, such that {f , g } = π(df , dg )
is a Poisson bracket on local holomorphic functions.
Symplectic leaves of a complex Poisson manifold (P, π) are the
integral submanifolds of the bundle map
π # : (T (1,0) P)∗ −→ T (1,0) P.
A T -Poisson manifold, where T is a complex torus, is a Poisson
manifold (P, π) with a T -action preserving π.
A T -leaf of a T -Poisson manifold is a T -orbit of symplectic leaves:
[
TΣ =
tΣ,
t∈T
where Σ is a symplectic leaf of π in P.
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Some correspondences
Poisson variety (P, π) ⇐⇒ Quantum algebra A;
Symplectic leaves of π ⇐⇒ Poisson primitive ideals of A;
T -leaves of π ⇐⇒ Goodearl-Letzler partition of Spec(A) into
tori;
Zariski closures of T -leaves of π ⇐⇒ T -invariant Poisson
prime ideals of A.
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Section 2. Poisson CGLs and examples
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Definition of Poisson CGLs
Definition: Let T be a k-torus.
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Definition of Poisson CGLs
Definition: Let T be a k-torus.
A T -Poisson CGL is a polynomial Poisson algebra
(A = k[x1 , x2 , . . . , xn ], { , }) with a T -action by Poisson
algebra automorphisms, such that
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Definition of Poisson CGLs
Definition: Let T be a k-torus.
A T -Poisson CGL is a polynomial Poisson algebra
(A = k[x1 , x2 , . . . , xn ], { , }) with a T -action by Poisson
algebra automorphisms, such that
1) each xi is a weight vector with weight λi ∈ Homk (T , k? );
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Definition of Poisson CGLs
Definition: Let T be a k-torus.
A T -Poisson CGL is a polynomial Poisson algebra
(A = k[x1 , x2 , . . . , xn ], { , }) with a T -action by Poisson
algebra automorphisms, such that
1) each xi is a weight vector with weight λi ∈ Homk (T , k? );
2) there exist ξ1 , . . . , ξn ∈ t = Lie(T ) s.t.
{xi , xj } ∈ λi (ξj )xi xj + k[xi+1 , . . . , xn ], 1 ≤ i < j ≤ n,
and λi (ξi ) 6= 0 for all 1 ≤ i ≤ n.
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Definition of Poisson CGLs
Definition: Let T be a k-torus.
A T -Poisson CGL is a polynomial Poisson algebra
(A = k[x1 , x2 , . . . , xn ], { , }) with a T -action by Poisson
algebra automorphisms, such that
1) each xi is a weight vector with weight λi ∈ Homk (T , k? );
2) there exist ξ1 , . . . , ξn ∈ t = Lie(T ) s.t.
{xi , xj } ∈ λi (ξj )xi xj + k[xi+1 , . . . , xn ], 1 ≤ i < j ≤ n,
and λi (ξi ) 6= 0 for all 1 ≤ i ≤ n.
A T -Poisson CGL is said to be symmetric if
{xi , xj } ∈ λi (ξj )xi xj + k[xi+1 , . . . , xj−1 ], 1 ≤ i < j ≤ n.
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Some facts on Poisson CGLs
T -Poisson CGLs are special iterated Ore-Poisson polynomial
algebras: for i = 1, . . . , n,
{xi , a} = xi ξi (a) + δi (a),
a ∈ k[xi+1 , . . . , xn ],
where ξi ∈ Lie(T ), and δi ∈ Derk (k[xi+1 , . . . , xn ]).
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Some facts on Poisson CGLs
T -Poisson CGLs are special iterated Ore-Poisson polynomial
algebras: for i = 1, . . . , n,
{xi , a} = xi ξi (a) + δi (a),
a ∈ k[xi+1 , . . . , xn ],
where ξi ∈ Lie(T ), and δi ∈ Derk (k[xi+1 , . . . , xn ]).
Theorem [Goodearl-Launois, 2007]. Assume that char(k) = 0.
A T -Poisson CGL on A = k[x1 , x2 , . . . , xn ] can have at most
2n T -stable Poisson prime ideals.
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Some facts on Poisson CGLs
T -Poisson CGLs are special iterated Ore-Poisson polynomial
algebras: for i = 1, . . . , n,
{xi , a} = xi ξi (a) + δi (a),
a ∈ k[xi+1 , . . . , xn ],
where ξi ∈ Lie(T ), and δi ∈ Derk (k[xi+1 , . . . , xn ]).
Theorem [Goodearl-Launois, 2007]. Assume that char(k) = 0.
A T -Poisson CGL on A = k[x1 , x2 , . . . , xn ] can have at most
2n T -stable Poisson prime ideals.
Poisson CGLs are the semiclassical limits of quantum CGLs,
which have been studied intensively by algebraists.
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Some facts on Poisson CGLs
T -Poisson CGLs are special iterated Ore-Poisson polynomial
algebras: for i = 1, . . . , n,
{xi , a} = xi ξi (a) + δi (a),
a ∈ k[xi+1 , . . . , xn ],
where ξi ∈ Lie(T ), and δi ∈ Derk (k[xi+1 , . . . , xn ]).
Theorem [Goodearl-Launois, 2007]. Assume that char(k) = 0.
A T -Poisson CGL on A = k[x1 , x2 , . . . , xn ] can have at most
2n T -stable Poisson prime ideals.
Poisson CGLs are the semiclassical limits of quantum CGLs,
which have been studied intensively by algebraists.
Systematic theory on cluster algebras based on Poisson CGLs
and quantum CGLs have been developed by Goodearl and
Yakimov.
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Our construction
Theorem: [Elek-L. 2012, 2016]:
1 Associated to each triple (G , s, γ) one has a T -Poisson CGL
on C[x1 , x2 , . . . , xn ], where
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Our construction
Theorem: [Elek-L. 2012, 2016]:
1 Associated to each triple (G , s, γ) one has a T -Poisson CGL
on C[x1 , x2 , . . . , xn ], where
G : a connected complex semisimple Lie group, T a maximal
torus of G ;
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Our construction
Theorem: [Elek-L. 2012, 2016]:
1 Associated to each triple (G , s, γ) one has a T -Poisson CGL
on C[x1 , x2 , . . . , xn ], where
G : a connected complex semisimple Lie group, T a maximal
torus of G ;
s = (s1 , s2 , . . . , sn ): a sequence of simple reflections in the
Weyl group W of G ;
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Our construction
Theorem: [Elek-L. 2012, 2016]:
1 Associated to each triple (G , s, γ) one has a T -Poisson CGL
on C[x1 , x2 , . . . , xn ], where
G : a connected complex semisimple Lie group, T a maximal
torus of G ;
s = (s1 , s2 , . . . , sn ): a sequence of simple reflections in the
Weyl group W of G ;
γ = (γ1 , γ2 , . . . , γn ): a subexpression of s, i.e., γj = e or sj for
each 1 ≤ j ≤ n, where e is the identity element of W .
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Our construction
Theorem: [Elek-L. 2012, 2016]:
1 Associated to each triple (G , s, γ) one has a T -Poisson CGL
on C[x1 , x2 , . . . , xn ], where
G : a connected complex semisimple Lie group, T a maximal
torus of G ;
s = (s1 , s2 , . . . , sn ): a sequence of simple reflections in the
Weyl group W of G ;
γ = (γ1 , γ2 , . . . , γn ): a subexpression of s, i.e., γj = e or sj for
each 1 ≤ j ≤ n, where e is the identity element of W .
2
The Poisson CGL is symmetric if γ = s;
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Our construction
Theorem: [Elek-L. 2012, 2016]:
1 Associated to each triple (G , s, γ) one has a T -Poisson CGL
on C[x1 , x2 , . . . , xn ], where
G : a connected complex semisimple Lie group, T a maximal
torus of G ;
s = (s1 , s2 , . . . , sn ): a sequence of simple reflections in the
Weyl group W of G ;
γ = (γ1 , γ2 , . . . , γn ): a subexpression of s, i.e., γj = e or sj for
each 1 ≤ j ≤ n, where e is the identity element of W .
2
The Poisson CGL is symmetric if γ = s;
3
Balazs Elek has written a computer program that computes
the Poisson brackets for any triple (G , s, γ).
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Our construction
Theorem: [Elek-L. 2012, 2016]:
1 Associated to each triple (G , s, γ) one has a T -Poisson CGL
on C[x1 , x2 , . . . , xn ], where
G : a connected complex semisimple Lie group, T a maximal
torus of G ;
s = (s1 , s2 , . . . , sn ): a sequence of simple reflections in the
Weyl group W of G ;
γ = (γ1 , γ2 , . . . , γn ): a subexpression of s, i.e., γj = e or sj for
each 1 ≤ j ≤ n, where e is the identity element of W .
2
The Poisson CGL is symmetric if γ = s;
3
Balazs Elek has written a computer program that computes
the Poisson brackets for any triple (G , s, γ).
4
[Mouquin-L. 2016] The T -leaves of these Poisson CGLs are
described using combinatorics of the Weyl group W .
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Examples of Poisson CGLs
Example. G = G2 and γ = s = (s1 , s2 , s1 , s2 , s1 , s2 ) give the
symmetric Poisson CGL on A = [x1 , x2 , x3 , x4 , x5 , x6 ]:
{x1 , x2 } = −3x1 x2
{x1 , x3 } = −x1 x3 − 2x2
{x1 , x4 } = −6x32
{x1 , x5 } = x1 x5 − 4x3
{x1 , x6 } = 3x1 x6 − 6x5
{x2 , x3 } = −3x2 x3
{x2 , x4 } = −6x33 − 3x2 x4
{x2 , x5 } = −6x32
{x2 , x6 } = 3x2 x6 − 18x3 x5 + 6x4
{x3 , x4 } = −3x3 x4 ,
{x3 , x6 } =
{x4 , x6 } =
−6x52 ,
−6x53 −
{x3 , x5 } = −x3 x5 − 2x4
{x4 , x5 } = −3x4 x5
3x4 x6 ,
{x5 , x6 } = −3x5 x6 .
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Another example. For G = G2 , s = (s1 , s2 , s1 , s1 , s2 , s1 , s1 , s1 , s2 ),
γ = (s1 , s2 , e, s1 , s2 , e, s1 , s1 , e), get T -Poisson CGL on
A = k[x1 , x2 , x3 , x4 , x5 , x6 , x7 , x8 , x9 ]:
{x1 , x2 } = −3x1 x2
{x1 , x3 } = 2x2 x32 + x1 x3
{x1 , x4 } = −4x2 x3 x4 − x1 x4 − 2x2
{x1 , x5 } = −6x3 x43 − 6x2 x3 x5 − 6x42
{x1 , x6 } = 6x3 x42 x62 + 2x2 x3 x6 + 4x4 x62 − x1 x6
{x1 , x7 } = −12x3 x42 x6 x7 − 2x2 x3 x7 − 6x3 x42 − 8x4 x6 x7 + x1 x7 − 4x4
{x1 , x8 } = 12x3 x42 x6 x8 + 2x2 x3 x8 + 8x4 x6 x8 − x1 x8
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{x1 , x9 } = −6x3 x5 x63 x73 x83 x92 + 18x3 x4 x62 x73 x83 x92 + 18x3 x5 x63 x72 x82 x92
− 18x3 x5 x62 x72 x83 x92 − 54x3 x4 x62 x72 x82 x92 + 36x3 x4 x6 x72 x83 x92
+ 6x62 x73 x83 x92 − 18x3 x5 x63 x7 x8 x92 + 36x3 x5 x62 x7 x82 x92
− 18x3 x5 x6 x7 x83 x92 + 54x3 x4 x62 x7 x8 x92 − 72x3 x4 x6 x7 x82 x92
+ 18x3 x4 x7 x83 x92 − 18x62 x72 x82 x92 + 12x6 x72 x83 x92
+ 6x3 x5 x63 x92 − 18x3 x5 x62 x8 x92 + 18x3 x5 x6 x82 x92
− 6x3 x5 x83 x92 − 18x3 x4 x62 x92 + 36x3 x4 x6 x8 x92 − 18x3 x4 x82 x92
+ 18x62 x7 x8 x92 − 24x6 x7 x82 x92 + 6x7 x83 x92 − 18x3 x42 x6 x9
− 6x62 x92 + 12x6 x8 x92 − 6x82 x92 − 6x2 x3 x9 − 12x4 x6 x9
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{x2 , x3 } = 3x2 x3
{x2 , x4 } = −3x2 x4
{x2 , x5 } = −6x43 − 3x2 x5
{x2 , x6 } = 6x42 x62
{x2 , x7 } = −12x42 x6 x7 − 6x42
{x2 , x8 } = 12x42 x6 x8
{x2 , x9 } = −6x5 x63 x73 x83 x92 + 18x4 x62 x73 x83 x92 + 18x5 x63 x72 x82 x92
− 18x5 x62 x72 x83 x92 − 54x4 x62 x72 x82 x92 + 36x4 x6 x72 x83 x92
− 18x5 x63 x7 x8 x92 + 36x5 x62 x7 x82 x92 − 18x5 x6 x7 x83 x92
+ 54x4 x62 x7 x8 x92 − 72x4 x6 x7 x82 x92 + 18x4 x7 x83 x92
+ 6x5 x63 x92 − 18x5 x62 x8 x92 + 18x5 x6 x82 x92 − 6x5 x83 x92
− 18x4 x62 x92 + 36x4 x6 x8 x92 − 18x4 x82 x92 − 18x42 x6 x9
− 3x2 x9
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{x3 , x4 } = −2x3 x4
{x3 , x5 } = −3x3 x5
{x3 , x6 } = x3 x6
{x3 , x7 } = −x3 x7
{x3 , x8 } = x3 x8
{x3 , x9 } = −3x3 x9
{x4 , x5 } = −3x4 x5
{x4 , x6 } = 2x5 x62 + x4 x6
{x4 , x7 } = −4x5 x6 x7 − x4 x7 − 2x5
{x4 , x8 } = 4x5 x6 x8 + x4 x8
{x4 , x9 } = 6x6 x73 x83 x92 − 18x6 x72 x82 x92 + 6x72 x83 x92 + 18x6 x7 x8 x92
− 12x7 x82 x92 − 6x5 x6 x9 − 6x6 x92 + 6x8 x92 − 3x4 x9
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{x5 , x6 } = 3x5 x6
{x5 , x7 } = −3x5 x7
{x5 , x8 } = 3x5 x8
{x5 , x9 } = 6x73 x83 x92 − 18x72 x82 x92 + 18x7 x8 x92 − 6x5 x9 − 6x92
{x6 , x7 } = −2x6 x7
{x6 , x8 } = 2x6 x8
{x6 , x9 } = −3x6 x9
{x7 , x8 } = 2x7 x8 − 2
{x7 , x9 } = −3x7 x9
{x8 , x9 } = 3x8 x9
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Section 3. Geometrical setting: projective Poisson manifolds paved
by Poisson CGLs
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The standard complex semisimple Poisson Lie group
G a connected, complex semisimple Lie group, e.g
G = SL(n, C), SO(n, C), SP(n, C), ...
(B, B− ) a pair of opposite Borel subgroups.
h , ig a symmetric, g-invariant, non-degenerate bilinear form,
e.g the Killing form on g
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The standard complex semisimple Poisson Lie group
G a connected, complex semisimple Lie group, e.g
G = SL(n, C), SO(n, C), SP(n, C), ...
(B, B− ) a pair of opposite Borel subgroups.
h , ig a symmetric, g-invariant, non-degenerate bilinear form,
e.g the Killing form on g
Well-known fact: The choice of (B, B− ) and h , ig defines a
holomorphic multiplicative Poisson structure πst on G ,
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The standard complex semisimple Poisson Lie group
G a connected, complex semisimple Lie group, e.g
G = SL(n, C), SO(n, C), SP(n, C), ...
(B, B− ) a pair of opposite Borel subgroups.
h , ig a symmetric, g-invariant, non-degenerate bilinear form,
e.g the Killing form on g
Well-known fact: The choice of (B, B− ) and h , ig defines a
holomorphic multiplicative Poisson structure πst on G , making
(G , πst ) into a Poisson Lie group.
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The standard complex semisimple Poisson Lie group
G a connected, complex semisimple Lie group, e.g
G = SL(n, C), SO(n, C), SP(n, C), ...
(B, B− ) a pair of opposite Borel subgroups.
h , ig a symmetric, g-invariant, non-degenerate bilinear form,
e.g the Killing form on g
Well-known fact: The choice of (B, B− ) and h , ig defines a
holomorphic multiplicative Poisson structure πst on G , making
(G , πst ) into a Poisson Lie group.
Example: For G = SL(n, C), write matrices in G as x = (xij ). Then
{xij , xkl }πst = (sign(k − i) − sign(l − j)) xil xkj .
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The standard complex semisimple Poisson Lie group
T = B ∩ B− , a maximal torus in G
W = NG (T )/T , the Weyl group of (G , T )
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The standard complex semisimple Poisson Lie group
T = B ∩ B− , a maximal torus in G
W = NG (T )/T , the Weyl group of (G , T )
Well-known facts:
1
(G , πst ) is a T -Poisson manifold under left translation by T ;
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The standard complex semisimple Poisson Lie group
T = B ∩ B− , a maximal torus in G
W = NG (T )/T , the Weyl group of (G , T )
Well-known facts:
1
(G , πst ) is a T -Poisson manifold under left translation by T ;
2
The T -leaves of πst are the double Bruhat cells
G u,v = BuB ∩ B− vB− ,
u, v ∈ W .
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The standard complex semisimple Poisson Lie group
T = B ∩ B− , a maximal torus in G
W = NG (T )/T , the Weyl group of (G , T )
Well-known facts:
1
(G , πst ) is a T -Poisson manifold under left translation by T ;
2
The T -leaves of πst are the double Bruhat cells
G u,v = BuB ∩ B− vB− ,
3
u, v ∈ W .
Double Bruhat cells have been studied intensively and have
been the motivating examples for cluster algebras, total
positivity, etc.
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The standard complex semisimple Poisson Lie group
T = B ∩ B− , a maximal torus in G
W = NG (T )/T , the Weyl group of (G , T )
Well-known facts:
1
(G , πst ) is a T -Poisson manifold under left translation by T ;
2
The T -leaves of πst are the double Bruhat cells
G u,v = BuB ∩ B− vB− ,
u, v ∈ W .
3
Double Bruhat cells have been studied intensively and have
been the motivating examples for cluster algebras, total
positivity, etc.
4
The Poisson structure πst on G projects to a well-defined
Poisson structure π1 on the flag variety G /B;
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Poisson structure on Bruhat cells as Poisson CGLs
Well-known facts:
1
T -leaves of (G /B, π1 ) are the open Richardson varieties
(BuB/B) ∩ (B− vB/B),
u, v ∈ W .
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Poisson structure on Bruhat cells as Poisson CGLs
Well-known facts:
1
T -leaves of (G /B, π1 ) are the open Richardson varieties
(BuB/B) ∩ (B− vB/B),
2
u, v ∈ W .
Every Bruhat cell BuB/B ⊂ G /B, where u ∈ W , is a Poisson
submanifold of (G /B, π1 );
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Poisson structure on Bruhat cells as Poisson CGLs
Well-known facts:
1
T -leaves of (G /B, π1 ) are the open Richardson varieties
(BuB/B) ∩ (B− vB/B),
2
u, v ∈ W .
Every Bruhat cell BuB/B ⊂ G /B, where u ∈ W , is a Poisson
submanifold of (G /B, π1 );
Theorem [Goodearl-Yakimov]: For u ∈ W , a reduced word
u = s1 s2 · · · sn of u gives rise to a parametrization BuB/B ∼
= Cn ,
with respect to which π1 is a symmetric T -Poisson CGL.
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Poisson structure on Bruhat cells as Poisson CGLs
Well-known facts:
1
T -leaves of (G /B, π1 ) are the open Richardson varieties
(BuB/B) ∩ (B− vB/B),
2
u, v ∈ W .
Every Bruhat cell BuB/B ⊂ G /B, where u ∈ W , is a Poisson
submanifold of (G /B, π1 );
Theorem [Goodearl-Yakimov]: For u ∈ W , a reduced word
u = s1 s2 · · · sn of u gives rise to a parametrization BuB/B ∼
= Cn ,
with respect to which π1 is a symmetric T -Poisson CGL.
Consequently, the Bruhat decomposition
G
G /B =
BuB/B
u∈W
is a paving of the projective Poisson variety (G /B, π1 ) by
symmetric T -Poisson CGLs.
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The Poisson manifold Fn = G ×B G ×B · · · ×B G /B
More generally, for any integer n ≥ 1, let
Fn = G ×B G ×B · · · ×B G /B
be the quotient of G n by B n by the right action
−1
(g1 , g2 , . . . , gn ) · (b1 , b2 , . . . , bn ) = (g1 b1 , b1−1 g2 b2 , . . . , bn−1
gn bn ),
and let $ : G n → Fn be the quotient map.
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The Poisson manifold Fn = G ×B G ×B · · · ×B G /B
More generally, for any integer n ≥ 1, let
Fn = G ×B G ×B · · · ×B G /B
be the quotient of G n by B n by the right action
−1
(g1 , g2 , . . . , gn ) · (b1 , b2 , . . . , bn ) = (g1 b1 , b1−1 g2 b2 , . . . , bn−1
gn bn ),
and let $ : G n → Fn be the quotient map.
!
n
z
}|
{
Lemma: The projection πn := $ πst × · · · × πst is a
well-defined Poisson structure on Fn .
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The Poisson manifold Fn = G ×B G ×B · · · ×B G /B
Theorem [Elek, Mouquin, L. 2016]
1 Have disjoint union decomposition
G
(Bu1 B ×B Bu2 B ×B · · · ×B Bun B/B),
Fn =
u=(u1 ,u2 ,...,un )∈W n
where each piece is a Poisson submanifold w.r.t. πn .
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The Poisson manifold Fn = G ×B G ×B · · · ×B G /B
Theorem [Elek, Mouquin, L. 2016]
1 Have disjoint union decomposition
G
(Bu1 B ×B Bu2 B ×B · · · ×B Bun B/B),
Fn =
u=(u1 ,u2 ,...,un )∈W n
where each piece is a Poisson submanifold w.r.t. πn .
2
Choices of reduced expressions for u1 , . . . , un give
parametrizations of
(Bu1 B ×B Bu2 B ×B · · · ×B Bun B/B) ∼
= Cl(u1 )+···l(un ) ,
w.r.t which πn defines a symmetric T -Poisson CGL.
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The Poisson manifold Fn = G ×B G ×B · · · ×B G /B
Theorem [Elek, Mouquin, L. 2016]
1 Have disjoint union decomposition
G
(Bu1 B ×B Bu2 B ×B · · · ×B Bun B/B),
Fn =
u=(u1 ,u2 ,...,un )∈W n
where each piece is a Poisson submanifold w.r.t. πn .
2
Choices of reduced expressions for u1 , . . . , un give
parametrizations of
(Bu1 B ×B Bu2 B ×B · · · ×B Bun B/B) ∼
= Cl(u1 )+···l(un ) ,
w.r.t which πn defines a symmetric T -Poisson CGL.
3
Consequently,
G
Fn =
(Bu1 B ×B Bu2 B ×B · · · ×B Bun B/B),
u=(u1 ,u2 ,...,un )∈W n
is a paving of the projective Poisson variety (Fn , πn ) by finitely
many Poisson CGLs.
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Section 4. Double Bruhat cells and symplectic groupoids
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Poisson and symplectic groupoids
Definition. A groupoid over a set X is a set Γ, together with
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Poisson and symplectic groupoids
Definition. A groupoid over a set X is a set Γ, together with
(1) source and target maps: α, β : Γ −→ X ;
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Poisson and symplectic groupoids
Definition. A groupoid over a set X is a set Γ, together with
(1) source and target maps: α, β : Γ −→ X ;
(2) multiplication: m : Γ2 −→ Γ, where
Γ2 := {(x, y ) ∈ Γ × Γ|β(x) = α(y )};
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Poisson and symplectic groupoids
Definition. A groupoid over a set X is a set Γ, together with
(1) source and target maps: α, β : Γ −→ X ;
(2) multiplication: m : Γ2 −→ Γ, where
Γ2 := {(x, y ) ∈ Γ × Γ|β(x) = α(y )};
(3) unit map : X −→ Γ;
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Poisson and symplectic groupoids
Definition. A groupoid over a set X is a set Γ, together with
(1) source and target maps: α, β : Γ −→ X ;
(2) multiplication: m : Γ2 −→ Γ, where
Γ2 := {(x, y ) ∈ Γ × Γ|β(x) = α(y )};
(3) unit map : X −→ Γ;
(4) inverse map: ι : Γ −→ Γ.
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Poisson and symplectic groupoids
Definition. A groupoid over a set X is a set Γ, together with
(1) source and target maps: α, β : Γ −→ X ;
(2) multiplication: m : Γ2 −→ Γ, where
Γ2 := {(x, y ) ∈ Γ × Γ|β(x) = α(y )};
(3) unit map : X −→ Γ;
(4) inverse map: ι : Γ −→ Γ. These maps must satisfy
m(m(x, y ), z) = m(x, m(y , z));
m((α(x)), x) = (x, (β(x))) = m(ι(x), x) = m(x, ι(x)) = x.
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Poisson and symplectic groupoids
Definition. A groupoid over a set X is a set Γ, together with
(1) source and target maps: α, β : Γ −→ X ;
(2) multiplication: m : Γ2 −→ Γ, where
Γ2 := {(x, y ) ∈ Γ × Γ|β(x) = α(y )};
(3) unit map : X −→ Γ;
(4) inverse map: ι : Γ −→ Γ. These maps must satisfy
m(m(x, y ), z) = m(x, m(y , z));
m((α(x)), x) = (x, (β(x))) = m(ι(x), x) = m(x, ι(x)) = x.
A Lie groupoid is a groupoid where all sets are manifolds and all
maps smooth.
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Poisson and symplectic groupoids
Definition.
A Poisson groupoid is a Lie groupoid Γ together with a
Poisson structure π such that the graph of the multiplication
is a coisotropic submanifold of (Γ × Γ × Γ, π × π × (−π)).
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Poisson and symplectic groupoids
Definition.
A Poisson groupoid is a Lie groupoid Γ together with a
Poisson structure π such that the graph of the multiplication
is a coisotropic submanifold of (Γ × Γ × Γ, π × π × (−π)).
A Poisson groupoid is a symplectic groupoid if the Poisson
structure is nondegenerate.
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Poisson and symplectic groupoids
Definition.
A Poisson groupoid is a Lie groupoid Γ together with a
Poisson structure π such that the graph of the multiplication
is a coisotropic submanifold of (Γ × Γ × Γ, π × π × (−π)).
A Poisson groupoid is a symplectic groupoid if the Poisson
structure is nondegenerate.
Theorem [Mouquin-L. 2016]
1 For any u ∈ W , the double Bruhat cell
G u,u = BuB ∩ B− uB− ⊂ G ,
equipped with the Poisson structure πst , is naturally a Poisson
groupoid over (BuB/B, π1 );
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Poisson and symplectic groupoids
Definition.
A Poisson groupoid is a Lie groupoid Γ together with a
Poisson structure π such that the graph of the multiplication
is a coisotropic submanifold of (Γ × Γ × Γ, π × π × (−π)).
A Poisson groupoid is a symplectic groupoid if the Poisson
structure is nondegenerate.
Theorem [Mouquin-L. 2016]
1 For any u ∈ W , the double Bruhat cell
G u,u = BuB ∩ B− uB− ⊂ G ,
equipped with the Poisson structure πst , is naturally a Poisson
groupoid over (BuB/B, π1 );
2
Symplectic leaves of πst in G u,u are symplectic groupoids over
(BuB/B, π1 ).
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The groupoid structure
The groupoid structure on G u,u : choose ū ∈ NG (T ) such that
u = ūT . Let Eū = N ū ∩ ūN− , where N and N− are the unipotent
radical of B and B− . Then have the diffeomorphisms
Eū × B −→ BuB,
(e, b) 7→ eb
B− × Eu −→ B− uB− ,
(b− , e) 7→ b− e.
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The groupoid structure
The groupoid structure on G u,u : choose ū ∈ NG (T ) such that
u = ūT . Let Eū = N ū ∩ ūN− , where N and N− are the unipotent
radical of B and B− . Then have the diffeomorphisms
Eū × B −→ BuB,
(e, b) 7→ eb
B− × Eu −→ B− uB− ,
(b− , e) 7→ b− e.
Let x = eb = b− e 0 ∈ G u,u = BuB ∩ B− uB− .
source map : sū (x) = x.B ∈ BuB/B ⊂ G /B
target map : tū (x) = e 0 .B ∈ BuB/B ⊂ G /B
identity bisection : Eū ⊂ G u,u
−1
e ∈ G u,u
inverse map : ιū (x) = e 0 b −1 = b−
0 00
multiplication: let y ∈ G u,u with tū (x) = sū (y ), i.ey = e 0 b 0 = b−
e .
0 e 00 ∈ G u,u .
Then x ? y = ebb 0 = b− b−
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References
J.-H. Lu and V. Mouquin, On the T -leaves of some Poisson
structures related to products of flag varieties,
arXiv:1511.02559.
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References
J.-H. Lu and V. Mouquin, On the T -leaves of some Poisson
structures related to products of flag varieties,
arXiv:1511.02559.
B. Elek and J.-H. Lu, On a Poisson structure on
Bott-Samelson varieties: computations in coordinates,
arXiv:1601.00047.
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References
J.-H. Lu and V. Mouquin, On the T -leaves of some Poisson
structures related to products of flag varieties,
arXiv:1511.02559.
B. Elek and J.-H. Lu, On a Poisson structure on
Bott-Samelson varieties: computations in coordinates,
arXiv:1601.00047.
J.-H. Lu and V. Mouquin, Double Bruhat cells and symplectic
gorupoids, arXiv:1607.00527.
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