CMUC, Univ. of Coimbra joint work with Joana Nunes da Costa web/Antunes.pdf ·...

Hypersymplectic structures on Courant algebroids

Paulo Antunes

CMUC, Univ. of Coimbra

joint work with Joana Nunes da Costa

XXI IFWGP, Burgos, 30/08/12

Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 1 / 33

Outline

1 Hypersymplectic structures on Lie algebroidsBasics on Lie algebroidsSymplectic structuresHypersymplectic structuresRelation with (para-)hyperkähler structuresExamples in T (R4)

2 Hypersymplectic structures on Courant algebroidsDefinition and examplesPropertiesRelation with (para-)hyperkähler structures

3 Some references

Hypersymplectic structures on Lie algebroids

Outline

3 Some references

Hypersymplectic structures on Lie algebroids Basics on Lie algebroids

Definition

Consider M a smooth manifold and A τ−→ Ma vector bundle.

Aρ //

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MA Lie algebroid structure on A τ−→ M is a pair (ρ, [., .]) where

• the anchor ρ : A −→ TM is a morphism of vector bundles,

• the bracket [., .] turns the space of sections Γ(A) into a Lie algebra,

such that the Leibniz rule

[X , fY ] = f [X ,Y ] + (ρ(X ) · f )Y

is satisfied for every f ∈ C∞(M) and X ,Y ∈ Γ(A).

Examples

1) Tangent bundle TMTM

C TMτ

}}{{{{

2) Lie algebra g

gρ≡0 //

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}}{{{{

3) Cotangent bundle T ∗M of a Poisson manifold(M, π), equipped with the bracket

[α, β]π

= Lπ#(α)β − Lπ#(β)α− d(π(α, β)),

for all α, β ∈ Γ(T ∗M).

T ∗Mπ] //

}}{{{{

Examples

C TMτ

}}{{{{

2) Lie algebra g

gρ≡0 //

��???

}}{{{{

[α, β]π

= Lπ#(α)β − Lπ#(β)α− d(π(α, β)),

T ∗Mπ] //

}}{{{{

Examples

C TMτ

}}{{{{

2) Lie algebra g

gρ≡0 //

��???

}}{{{{

[α, β]π

= Lπ#(α)β − Lπ#(β)α− d(π(α, β)),

T ∗Mπ] //

}}{{{{

Nijenhuis tensor

(A, ρ, [·, ·]) Lie algebroid over M and N ∈ Γ(A⊗ A∗) a (1, 1)-tensor seen asa bundle endomorphism N : A→ A.

Nijenhuis torsion of N:

TN(X ,Y ) = [NX ,NY ]− N ([NX ,Y ] + [X ,NY ]− N[X ,Y ])︸︷︷︸[X ,Y ]N

= [NX ,NY ]− N[X ,Y ]N

or, equivalently,

TN(X ,Y ) =12

(([X ,Y ]N

)N − [X ,Y ]N2

N is a Nijenhuis tensor if TN = 0.

Nijenhuis tensor

= [NX ,NY ]− N[X ,Y ]N

or, equivalently,

TN(X ,Y ) =12

(([X ,Y ]N

)N − [X ,Y ]N2

Nijenhuis tensor

= [NX ,NY ]− N[X ,Y ]N

or, equivalently,

TN(X ,Y ) =12

(([X ,Y ]N

)N − [X ,Y ]N2

Nijenhuis tensor

= [NX ,NY ]− N[X ,Y ]N

or, equivalently,

TN(X ,Y ) =12

(([X ,Y ]N

)N − [X ,Y ]N2

(Para-)Complex tensors

(A, ρ, [·, ·]) Lie algebroid over M and I ∈ Γ(A⊗ A∗) a (1, 1)-tensor seen asa bundle endomorphism I : A→ A.

The (1, 1)-tensor I ∈ Γ(A⊗ A∗) is a complex tensor if it satisfies{I 2 = −IdA;T I = 0.

The (1, 1)-tensor I ∈ Γ(A⊗ A∗) is a para-complex tensor if it satisfies{I 2 = IdA;T I = 0.

Exterior differential

DEFINITIONThe exterior differential d : Γ(∧•(A∗)) → Γ(∧•+1(A∗)) is defined by setting for allη ∈ Γ(∧k(A∗)),

dη(X0, . . . ,Xk) :=k∑

(−1)i ρ(Xi ) · η(X0, . . . , X̂i , . . . ,Xk)

0≤i<j≤k

(−1)i+j η([

Xi ,Xj],X0, . . . , X̂i , . . . , X̂j , . . . ,Xk

for all X0, . . . ,Xk ∈ Γ(A), where the symbol ̂ means that the term below is omitted.

Examples1 If A = TM (and ρ = Id), then d is the de Rham differential.

2 If A = g is a Lie algebra, then d is the Chevalley-Eilenberg differential.

3 If A = T ∗M, with (M, π) a Poisson manifold, then d is the Lichnerowiczdifferential, d(.) = [π, .]

Exterior differential

DEFINITIONThe exterior differential d : Γ(∧•(A∗)) → Γ(∧•+1(A∗)) is defined by setting for allη ∈ Γ(∧k(A∗)),

dη(X0, . . . ,Xk) :=k∑

(−1)i ρ(Xi ) · η(X0, . . . , X̂i , . . . ,Xk)

0≤i<j≤k

(−1)i+j η([

Xi ,Xj],X0, . . . , X̂i , . . . , X̂j , . . . ,Xk

for all X0, . . . ,Xk ∈ Γ(A), where the symbol ̂ means that the term below is omitted.

Examples1 If A = TM (and ρ = Id), then d is the de Rham differential.

2 If A = g is a Lie algebra, then d is the Chevalley-Eilenberg differential.

3 If A = T ∗M, with (M, π) a Poisson manifold, then d is the Lichnerowiczdifferential, d(.) = [π, .]

Hypersymplectic structures on Lie algebroids Symplectic structures

Symplectic structures

DEFINITIONOn a Lie algebroid (A, ρ, [·, ·]), a symplectic structure is a sectionω ∈ Γ(∧2A∗) which is

non-degenerate, i.e., ∃π ∈ Γ(∧2A) such that π] ◦ ω[ = IdA;closed, i.e, such that dω = 0.

In the above definition, we used the morphisms π] and ω[ defined asfollows:

π] : Γ(A∗)→ Γ(A) ω[ : Γ(A)→ Γ(A∗)

〈β, π](α)〉 := π(α, β) 〈ω[(X ),Y 〉 := ω(X ,Y )

Hypersymplectic structures on Lie algebroids Hypersymplectic structures

Hypersymplectic structures

Consider 3 symplectic forms ω1, ω2 andω3 ∈ Γ(∧2A∗) with inverse Poisson bivec-tors π1, π2 and π3 ∈ Γ(∧2A), respectively.We define the transition (1, 1)-tensorsI1, I2, I3 : Γ(A)→ Γ(A), by setting

Ij := π]j−1 ◦ ω[j+1.

considering the indices as elements of Z3.

The triple (ω1, ω2, ω3) is an ε-hypersymplectic structure on the Liealgebroid (A, ρ, [·, ·]) if the transition (1, 1)-tensors satisfies

Ij2 = εj IdA,

where the parameters εj = ±1 form the triple ε = (ε1, ε2, ε3).

Hypersymplectic structures

Consider 3 symplectic forms ω1, ω2 andω3 ∈ Γ(∧2A∗) with inverse Poisson bivec-tors π1, π2 and π3 ∈ Γ(∧2A), respectively.We define the transition (1, 1)-tensorsI1, I2, I3 : Γ(A)→ Γ(A), by setting

Ij := π]j−1 ◦ ω[j+1.

considering the indices as elements of Z3.

The triple (ω1, ω2, ω3) is an ε-hypersymplectic structure on the Liealgebroid (A, ρ, [·, ·]) if the transition (1, 1)-tensors satisfies

Ij2 = εj IdA,

where the parameters εj = ±1 form the triple ε = (ε1, ε2, ε3).

Transition (1, 1)-tensors

When (ω1, ω2, ω3) is an ε-hypersymplectic structure, the transition(1, 1)-tensors satisfy, for all j 6= k ∈ {1, 2, 3}

1 (Ij)−1 = εj Ij ;2 I3I2I1 = IdA and I1I2I3 = ε1ε2ε3 IdA;3 Ij = εj Ij−1Ij+1 = εj+1εj−1 Ij+1Ij−1;4 Ij Ik = ε1ε2ε3Ik Ij ;5 T Ij = 0.

In order to have anti-commuting transition (1, 1)-tensors, we consider thecase ε1ε2ε3 = −1.Then, we have to distinguish two cases:

ε1 = ε2 = ε3 = −1 → hypersymplectic structure;ε1 = ε2 = 1 and ε3 = −1 → para-hypersymplectic structure.

In order to have anti-commuting transition (1, 1)-tensors, we consider thecase ε1ε2ε3 = −1.

Then, we have to distinguish two cases:

ε1 = ε2 = ε3 = −1 → hypersymplectic structure;

ε1 = ε2 = 1 and ε3 = −1 → para-hypersymplectic structure.

(Candidate to) Induced pseudo-metricGiven an ε-hypersymplectic structure, we define a vector bundle morphism

g : A→ A∗

by setting

g := ε3ε2 ω3[ ◦ π1

] ◦ ω2[

= ε1ε3 ω1[ ◦ π2

] ◦ ω3[

= ε2ε1 ω2[ ◦ π3

] ◦ ω1[.

Notice that the definition of g is not affected by a circular permutation ofthe indices.

Then, we set as definition of g, for any i ∈ {1, 2, 3},

g := εi−1εi+1 ωi−1[ ◦ πi

] ◦ ωi+1[.

g : A→ A∗

by setting

g := ε3ε2 ω3[ ◦ π1

] ◦ ω2[

= ε1ε3 ω1[ ◦ π2

] ◦ ω3[

= ε2ε1 ω2[ ◦ π3

] ◦ ω1[.

g := εi−1εi+1 ωi−1[ ◦ πi

] ◦ ωi+1[.

g : A→ A∗

by setting

g := ε3ε2 ω3[ ◦ π1

] ◦ ω2[

= ε1ε3 ω1[ ◦ π2

] ◦ ω3[

= ε2ε1 ω2[ ◦ π3

] ◦ ω1[.

g := εi−1εi+1 ωi−1[ ◦ πi

] ◦ ωi+1[.

g : A→ A∗

by setting

g := ε3ε2 ω3[ ◦ π1

] ◦ ω2[

= ε1ε3 ω1[ ◦ π2

] ◦ ω3[

= ε2ε1 ω2[ ◦ π3

] ◦ ω1[.

Notice that the definition of g is not affected by a circular permutation ofthe indices.Then, we set as definition of g, for any i ∈ {1, 2, 3},

g := εi−1εi+1 ωi−1[ ◦ πi

] ◦ ωi+1[.

Induced pseudo-metricConsider (ω1, ω2, ω3) a (para-)hypersymplectic structure, i.e., such thatε1ε2ε3 = −1. Then the morphism g : Γ(A)→ Γ(A∗) is a pseudo-metric onA→ M, i.e.,

C∞(M)-linear;symmetric;non-degenerate.

We can remove the “pseudo” prefix if g is positive definite, i.e., if it satisfies

〈g(X ),X 〉 > 0,

for all non vanishing sections X ∈ Γ(A).

In what follows, we do not ask for the metric to be positive definite.However, in order to simplify the terminology we will omit the “pseudo”prefix.

〈g(X ),X 〉 > 0,

Hypersymplectic structures on Lie algebroids Relation with (para-)hyperkähler structures

Relation with (para-)hyperkählerDEFINITIONA pair (g, I ) is a para-hermitian structure if the following is satisfied

g is a metric;I is a para-complex tensor;〈g(IX ), IY 〉 = −〈g(X ),Y 〉, for all X ,Y ∈ Γ(A).

DEFINITIONA quadruple (I1, I2, I3, g) is a para-hyperkähler structure if the following issatisfied

g is a metric;I1, I2 are anti-commuting para-complex tensors and I3 = I1I2;(g, Ij)j=1,2 are para-hermitian structures;ωj[ = g ◦ Ij are closed 2-forms.

Relation with (para-)hyperkählerDEFINITIONA pair (g, I ) is a para-hermitian structure if the following is satisfied

g is a metric;I is a para-complex tensor;〈g(IX ), IY 〉 = −〈g(X ),Y 〉, for all X ,Y ∈ Γ(A).

DEFINITIONA quadruple (I1, I2, I3, g) is a para-hyperkähler structure if the following issatisfied

g is a metric;I1, I2 are anti-commuting para-complex tensors and I3 = I1I2;(g, Ij)j=1,2 are para-hermitian structures;ωj[ = g ◦ Ij are closed 2-forms.

Relation with (para-)hyperkähler

THEOREMIf (ω1, ω2, ω3) is a para-hypersymplectic structure then (I1, I2, I3, g) is apara-hyperkähler structure.

Induced structures and vice-versa

We defined the morphisms g, Ij , j = 1, 2, 3, starting from anε-hypersymplectic structure (ω1, ω2, ω3).

But the process can be reversed.

In fact, we have

ωj[ = εjεj−1 g ◦ Ij

In fact, we have

1-1 correspondence

THEOREMThe triple (ω1, ω2, ω3) is a para-hypersymplectic structure if and only if(I1, I2, I3, g) is a para-hyperkähler structure.

Hypersymplectic structures on Lie algebroids Examples in T (R4)

Examples in T (R4)

Consider in R4 the coordinates (x , y , p, q) and the following six 2-forms inR4

ω1 = dx ∧ dp + dy ∧ dq; ω4 = dx ∧ dp − dy ∧ dq;

ω2 = dx ∧ dq + dp ∧ dy ; ω5 = dx ∧ dq − dp ∧ dy ;

ω3 = dx ∧ dy − dp ∧ dq; ω6 = dx ∧ dy + dp ∧ dq.

These 2-forms on R4 are symplectic and form a basis of the vector space ofsections Γ(∧2(T ∗R4)).

Examples in T (R4)

Consider in R4 the coordinates (x , y , p, q) and the following six 2-forms inR4

ω1 = dx ∧ dp + dy ∧ dq; ω4 = dx ∧ dp − dy ∧ dq;

ω2 = dx ∧ dq + dp ∧ dy ; ω5 = dx ∧ dq − dp ∧ dy ;

ω3 = dx ∧ dy − dp ∧ dq; ω6 = dx ∧ dy + dp ∧ dq.

These 2-forms on R4 are symplectic and form a basis of the vector space ofsections Γ(∧2(T ∗R4)).

Examples in T (R4)

For all i 6= j 6= k ∈ {1, . . . 6}, the triple (ωi , ωj , ωk) is a(para-)hypersymplectic structure on the Lie algebroid T (R4). Moreprecisely,

1 The triples (ω1, ω2, ω3) and(ω4, ω5, ω6) are hypersymplecticstructures.

2 The 9 triples (ωi , ωj , ωk) where1 ≤ i ≤ j ≤ 3 and k ∈ {4, 5, 6} arepara-hypersymplectic structures.

Examples in T (R4)

Hypersymplectic structures on Courant algebroids

Outline

3 Some references

Hypersymplectic structures on Courant algebroids Definition and examples

Courant algebroids

Courant algebroid over M, (E , ρ, 〈·, ·〉, [·, ·]):

Eρ //

}}zzzz

〈·, ·〉 symmetric non-degenerate bilinear form on E[·, ·] Loday bracket on Γ(E ), i.e.,R-bilinear and [X , [Y ,Z ]] = [[X ,Y ],Z ] + [Y , [X ,Z ]]

ρ(X ).〈Y ,Z 〉 = 〈[X ,Y ],Z 〉+ 〈Y , [X ,Z ]〉ρ(X ).〈Y ,Z 〉 = 〈X , [Y ,Z ]〉+ 〈X , [Z ,Y ]〉

Courant algebroids

Eρ //

}}zzzz

Courant algebroids

Eρ //

}}zzzz

〈·, ·〉 symmetric non-degenerate bilinear form on E

[·, ·] Loday bracket on Γ(E ), i.e.,R-bilinear and [X , [Y ,Z ]] = [[X ,Y ],Z ] + [Y , [X ,Z ]]

Courant algebroids

Eρ //

}}zzzz

Courant algebroids

Eρ //

}}zzzz

ρ(X ).〈Y ,Z 〉 = 〈[X ,Y ],Z 〉+ 〈Y , [X ,Z ]〉

ρ(X ).〈Y ,Z 〉 = 〈X , [Y ,Z ]〉+ 〈X , [Z ,Y ]〉

Courant algebroids

Eρ //

}}zzzz

Examples of Courant algebroids

Generalized tangent bundle TM ⊕ T ∗M

TM ⊕ T ∗MIdTM ⊕ 0 //

%%LLLLLLLLLL TM

}}{{{{

I 〈X + α,Y + β〉 = iXβ + iYαI [X + α,Y + β] = [X ,Y ] + (LXβ − iY dα)

Double of a Lie (bi)algebroid A⊕ A∗

%%LLLLLLLLLL TM

}}{{{{

I 〈X + α,Y + β〉 = iXβ + iYα

I [X + α,Y + β] = [X ,Y ] + (LXβ − iY dα)

%%LLLLLLLLLL TM

}}{{{{

%%LLLLLLLLLL TM

}}{{{{

Definition

(E , ρ, 〈·, ·〉, [·, ·]) a Courant algebroid over M.

DEFINITIONAn ε-hypersymplectic structure on E is a triple (S1,S2,S3) ofskew-symmetric bundle maps Si : E → E, i ∈ {1, 2, 3}, such that

1 Si2 = εi IdE ,

2 TSi = 0,where the parameters εi = ±1 form the triple ε = (ε1, ε2, ε3). Moreoverthe bundle maps (Si )i=1,2,3 ε-commute in the sense that, for alli 6= j ∈ {1, 2, 3},

3 SiSj = ε1ε2ε3 SjSi .

Definition

(E , ρ, 〈·, ·〉, [·, ·]) a Courant algebroid over M.

DEFINITIONAn ε-hypersymplectic structure on E is a triple (S1,S2,S3) ofskew-symmetric bundle maps Si : E → E, i ∈ {1, 2, 3}, such that

1 Si2 = εi IdE ,

2 TSi = 0,where the parameters εi = ±1 form the triple ε = (ε1, ε2, ε3). Moreoverthe bundle maps (Si )i=1,2,3 ε-commute in the sense that, for alli 6= j ∈ {1, 2, 3},

3 SiSj = ε1ε2ε3 SjSi .

Example on (A⊕ A∗)

Consider (ω1, ω2, ω3) an ε-hypersymplectic structure on the Lie algebroid(A, ρ, [·, ·]).

For each i ∈ {1, 2, 3}, let us define a bundle endomorphismSi : A⊕ A∗ → A⊕ A∗ given in a matrix form by

[0 εi πiωi 0

The morphisms Si satisfy the following

I Si2 = εi IdA⊕A∗ I TSi = 0 I SiSj = ε1ε2ε3 SjSi

The triple (S1,S2,S3) is an ε-hypersymplectic structure on the Courantalgebroid A⊕ A∗.

[0 εi πiωi 0

Induced transition tensors and metric

Consider (S1,S2,S3) an ε-hypersymplectic structure on the Courantalgebroid (E , ρ, 〈·, ·〉, [·, ·]).

For each i ∈ {1, 2, 3}, let us define the tran-sition tensors Ti : E → E by setting

Ti := εi−1Si−1Si+1.

and the (wannabe) metric G : E → E ∗ by setting

G := Si+1SiSi−1.

Continued example on (A⊕ A∗)

Start from an ε-hypersymplectic structure (ω1, ω2, ω3) on the Lie algebroid(A, ρ, [·, ·]) and build the matrices

[0 εi πiωi 0

I The respective transition tensors are given, in matrix form, by

[Ii 00 ε1ε2ε3 Ii ∗

]I The (wannabe) metric is given, in matrix form, by

[0 g−1

[0 εi πiωi 0

[Ii 00 ε1ε2ε3 Ii ∗

I The (wannabe) metric is given, in matrix form, by

[0 g−1

[0 εi πiωi 0

[Ii 00 ε1ε2ε3 Ii ∗

]I The (wannabe) metric is given, in matrix form, by

[0 g−1

Hypersymplectic structures on Courant algebroids Properties

Properties

Consider (S1,S2,S3) an ε-hypersymplectic structure on the Courantalgebroid (E , ρ, 〈·, ·〉, [·, ·]).The transition tensors satisfy

I Ti 2 = εi IdE ; I TiTj = ε1ε2ε3TjTi ;I Ti ∗ = ε1ε2ε3Ti ; I T3T2T1 = IdE .

I As in Lie algebroids, in order to have objects with nice properties to workwith (skew-symmetric, anti-commuting, vanishing Nijenhuis torsion. . . ) wewill restrict our study to the case where ε1ε2ε3 = −1.

I Then, we have to distinguish two cases:

ε1 = ε2 = ε3 = −1 → hypersymplectic structure on E ;ε1 = ε2 = 1 and ε3 = −1 → para-hypersymplectic structure on E .

Properties

I Then, we have to distinguish two cases:ε1 = ε2 = ε3 = −1 → hypersymplectic structure on E ;

ε1 = ε2 = 1 and ε3 = −1 → para-hypersymplectic structure on E .

Properties

Transition tensors are integrable

PROPOSITIONIf (S1,S2,S3) is a (para-)hypersymplectic structure, then Ti is a{

complex tensor, if εi = −1;para-complex tensor, if εi = 1.

Proof.If I , J are anticommuting Nijenhuis tensors on E then I ◦ J is a Nijenhuistensor on E .

Transition tensors are integrable

PROPOSITIONIf (S1,S2,S3) is a (para-)hypersymplectic structure, then Ti is a{

complex tensor, if εi = −1;para-complex tensor, if εi = 1.

Proof.If I , J are anticommuting Nijenhuis tensors on E then I ◦ J is a Nijenhuistensor on E .

G is in fact a metric

DEFINITIONA metric on (E , ρ, 〈·, ·〉, [·, ·]) is an orthogonal and symmetric bundleautomorphism G : E → E.

PROPOSITIONIf (S1,S2,S3) is a (para-)hypersymplectic structure, then G is a metric.

G is in fact a metric

DEFINITIONA metric on (E , ρ, 〈·, ·〉, [·, ·]) is an orthogonal and symmetric bundleautomorphism G : E → E.

PROPOSITIONIf (S1,S2,S3) is a (para-)hypersymplectic structure, then G is a metric.

Hypersymplectic structures on Courant algebroids Relation with (para-)hyperkähler structures

Relation with (para-)hyperkähler structures

(E , ρ, 〈·, ·〉, [·, ·]) a Courant algebroid.

DEFINITIONA pair (G, T ) is a para-hermitian structure if the following is satisfied

G is a metricT is a para-complex tensor〈G(T X ), T Y 〉 = −〈G(X ),Y 〉, for all X ,Y ∈ Γ(E ).

PROPOSITIONWhen (S1,S2,S3) is a para-hypersymplectic structure, the pairs(G, Tj)j=1,2 are para-hermitian structures (and the pair (G, T3) is always anhermitian structure).

Relation with (para-)hyperkähler structures

(E , ρ, 〈·, ·〉, [·, ·]) a Courant algebroid.

DEFINITIONA pair (G, T ) is a para-hermitian structure if the following is satisfied

G is a metricT is a para-complex tensor〈G(T X ), T Y 〉 = −〈G(X ),Y 〉, for all X ,Y ∈ Γ(E ).

PROPOSITIONWhen (S1,S2,S3) is a para-hypersymplectic structure, the pairs(G, Tj)j=1,2 are para-hermitian structures (and the pair (G, T3) is always anhermitian structure).

Relation with para-hyperkähler

DEFINITIONA quadruple (T1, T2, T3,G) is a para-hyperkähler structure if the following issatisfied

G is a metricT1, T2 are anti-commuting para-complex tensors and T3 = T1T2(G, Tj)j=1,2 are para-hermitian structuresT(GTj) = 0, j = 1, 2, 3.

THEOREMThe triple (S1,S2,S3) is a para-hypersymplectic structure iff (T1, T2, T3,G)is a para-hyperkähler structure.

Relation with para-hyperkähler

DEFINITIONA quadruple (T1, T2, T3,G) is a para-hyperkähler structure if the following issatisfied

G is a metricT1, T2 are anti-commuting para-complex tensors and T3 = T1T2(G, Tj)j=1,2 are para-hermitian structuresT(GTj) = 0, j = 1, 2, 3.

THEOREMThe triple (S1,S2,S3) is a para-hypersymplectic structure iff (T1, T2, T3,G)is a para-hyperkähler structure.

Some references

Outline

3 Some references

Some references

Some referencesP. Xu, Hyper-Lie Poisson structures, Annales Scientifiques de l’ÉcoleNormale Supérieure 30, Issue 3, 279–302 (1997).P. Antunes, Crochets de Poisson gradués et applications: structurescompatibles et généralisations des structures hyperkählériennes, Thèsede doctorat de l’École Polytechnique, (2010).H. Bursztyn, G. Cavalcanti and M. Gualtieri, Generalized Kähler andhyper-Kähler quotients, Poisson geometry in mathematics and physics,Contemp. Math., 450, 61–77, Amer. Math. Soc., Providence, RI,(2008).N. J. Hitchin, Hypersymplectic quotients, La “Mécanique analytique”de Lagrange et son héritage, Atti della Accademia delle Scienze diTorino, Classe de Scienze fisiche Matematiche e Naturali, Suplementoal numero 124, 1990, 169–180.P. Antunes, C. Laurent-Gengoux, J. M. Nunes da Costa, Hierarchiesand compatibility on Courant algebroids, arXiv:1111.0800.

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