Generalized Hypersymplectic Structures Paulo Antunes CMUC, Univ. of - - PowerPoint PPT Presentation

Generalized Hypersymplectic Structures

Paulo Antunes

CMUC, Univ. of Coimbra joint work with Joana Nunes da Costa

QUISG, Olhão, 06/09/12

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 1 / 35

Outline

1

Hypersymplectic structures on Lie algebroids Basics on Lie algebroids Symplectic structures Hypersymplectic structures Relation with (para-)hyperkähler structures Examples in T(R4)

2

Hypersymplectic structures on Courant algebroids Definition and examples Properties Relation with (para-)hyperkähler structures

3

Some references

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 2 / 35

Hypersymplectic structures on Lie algebroids

Outline

1

Hypersymplectic structures on Lie algebroids Basics on Lie algebroids Symplectic structures Hypersymplectic structures Relation with (para-)hyperkähler structures Examples in T(R4)

2

Hypersymplectic structures on Courant algebroids Definition and examples Properties Relation with (para-)hyperkähler structures

3

Some references

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 3 / 35

Hypersymplectic structures on Lie algebroids Basics on Lie algebroids

Definition

Consider M a smooth manifold and A

τ

− → M a vector bundle. A

ρ

• τ
• TM
• M

A 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).

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 4 / 35

Hypersymplectic structures on Lie algebroids Basics on Lie algebroids

Examples

1) Tangent bundle TM TM

id

• τ
• TM

τ

• M

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 5 / 35

Hypersymplectic structures on Lie algebroids Basics on Lie algebroids

Examples

1) Tangent bundle TM TM

id

• τ
• TM

τ

• M

2) Lie algebra g g

ρ≡0

• {0}
• {∗}

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 5 / 35

Hypersymplectic structures on Lie algebroids Basics on Lie algebroids

Examples

1) Tangent bundle TM TM

id

• τ
• TM

τ

• M

2) Lie algebra g g

ρ≡0

• {0}
• {∗}

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 ∗M

π♯

• TM
• M

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 5 / 35

Hypersymplectic structures on Lie algebroids Basics on Lie algebroids

Nijenhuis tensor

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

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 6 / 35

Hypersymplectic structures on Lie algebroids Basics on Lie algebroids

Nijenhuis tensor

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

T I(X, Y ) = [IX, IY ] − I ([IX, Y ] + [X, IY ] − I[X, Y ])

• [X,Y ]I

= [IX, IY ] − I[X, Y ]I

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 6 / 35

Hypersymplectic structures on Lie algebroids Basics on Lie algebroids

Nijenhuis tensor

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

T I(X, Y ) = [IX, IY ] − I ([IX, Y ] + [X, IY ] − I[X, Y ])

• [X,Y ]I

= [IX, IY ] − I[X, Y ]I

• r, equivalently,

T I(X, Y ) = 1

2

• [X, Y ]I
• I − [X, Y ]I 2
• .

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 6 / 35

Hypersymplectic structures on Lie algebroids Basics on Lie algebroids

Nijenhuis tensor

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

T I(X, Y ) = [IX, IY ] − I ([IX, Y ] + [X, IY ] − I[X, Y ])

• [X,Y ]I

= [IX, IY ] − I[X, Y ]I

• r, equivalently,

T I(X, Y ) = 1

2

• [X, Y ]I
• I − [X, Y ]I 2
• .

I is a Nijenhuis tensor if T I = 0.

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 6 / 35

Hypersymplectic structures on Lie algebroids Basics on Lie algebroids

(Para-)Complex tensors

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

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 7 / 35

Hypersymplectic structures on Lie algebroids Basics on Lie algebroids

(Para-)Complex tensors

(A, ρ, [·, ·]) Lie algebroid over M I ∈ Γ(A ⊗ A∗) a (1, 1)-tensor seen as a 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.

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 7 / 35

Hypersymplectic structures on Lie algebroids Basics on Lie algebroids

(Para-)Complex tensors

(A, ρ, [·, ·]) Lie algebroid over M I ∈ Γ(A ⊗ A∗) a (1, 1)-tensor seen as a 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.

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 7 / 35

Hypersymplectic structures on Lie algebroids Basics on Lie algebroids

Exterior differential

DEFINITION

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

dη(X0, . . . , Xk) :=

k

• i=0

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

• 0≤i<j≤k

(−1)i+j η

• Xi, Xj
• , X0, . . . ,

Xi, . . . , Xj, . . . , Xk

• ,

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

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 8 / 35

Hypersymplectic structures on Lie algebroids Basics on Lie algebroids

Exterior differential

DEFINITION

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

dη(X0, . . . , Xk) :=

k

• i=0

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

• 0≤i<j≤k

(−1)i+j η

• Xi, Xj
• , X0, . . . ,

Xi, . . . , Xj, . . . , Xk

• ,

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

Examples

1

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 Lichnerowicz differential, d(.) = [π, .]

SN.

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 8 / 35

Hypersymplectic structures on Lie algebroids Symplectic structures

Symplectic structures

DEFINITION

On a Lie algebroid (A, ρ, [·, ·]), a symplectic structure is a section ω ∈ 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 as follows: π♯ : Γ(A∗) → Γ(A) ω♭ : Γ(A) → Γ(A∗) β, π♯(α) := π(α, β) ω♭(X), Y := ω(X, Y )

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 9 / 35

Hypersymplectic structures on Lie algebroids Hypersymplectic structures

Hypersymplectic structures

Consider 3 symplectic forms ω1, ω2 and ω3 ∈ Γ(∧2A∗) with inverse Poisson bivec- tors π1, π2 and π3 ∈ Γ(∧2A), respectively. We define the transition (1, 1)-tensors I1, I2, I3 : Γ(A) → Γ(A), by setting Ij := π♯

j−1 ◦ ω♭ j+1.

considering the indices as elements of Z3.

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 10 / 35

Hypersymplectic structures on Lie algebroids Hypersymplectic structures

Hypersymplectic structures

Consider 3 symplectic forms ω1, ω2 and ω3 ∈ Γ(∧2A∗) with inverse Poisson bivec- tors π1, π2 and π3 ∈ Γ(∧2A), respectively. We define the transition (1, 1)-tensors I1, 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 Lie algebroid (A, ρ, [·, ·]) if the transition (1, 1)-tensors satisfies Ij 2 = εj IdA, where the parameters εj = ±1 form the triple ε = (ε1, ε2, ε3).

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 10 / 35

Hypersymplectic structures on Lie algebroids Hypersymplectic structures

Transition (1, 1)-tensors

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

1 (Ij)−1 = εjIj; 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. Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 11 / 35

Hypersymplectic structures on Lie algebroids Hypersymplectic structures

Transition (1, 1)-tensors

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

1 (Ij)−1 = εjIj; 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 the case ε1ε2ε3 = −1.

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 11 / 35

Hypersymplectic structures on Lie algebroids Hypersymplectic structures

Transition (1, 1)-tensors

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

1 (Ij)−1 = εjIj; 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 the case ε1ε2ε3 = −1. Then, we have to distinguish two cases: ε1 = ε2 = ε3 = −1 → hypersymplectic structure;

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 11 / 35

Hypersymplectic structures on Lie algebroids Hypersymplectic structures

Transition (1, 1)-tensors

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

1 (Ij)−1 = εjIj; 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 the case ε1ε2ε3 = −1. Then, we have to distinguish two cases: ε1 = ε2 = ε3 = −1 → hypersymplectic structure; ε1 = ε2 = 1 and ε3 = −1 → para-hypersymplectic structure.

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 11 / 35

Hypersymplectic structures on Lie algebroids Hypersymplectic structures

Transition (1, 1)-tensors

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

1 (Ij)−1 = εjIj; 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 the case ε1ε2ε3 = −1. Then, we have to distinguish two cases: ε1 = ε2 = ε3 = −1 → hypersymplectic structure; ε1 = ε2 = 1 and ε3 = −1 → para-hypersymplectic structure.

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 11 / 35

Hypersymplectic structures on Lie algebroids Hypersymplectic structures

(Candidate to) Induced pseudo-metric

Given an ε-hypersymplectic structure, we define a vector bundle morphism g : A → A∗ by setting g := ε3ε2 ω3♭ ◦ π1♯ ◦ ω2♭ Notice that the definition of g is not affected by a circular permutation of the indices.

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 12 / 35

Hypersymplectic structures on Lie algebroids Hypersymplectic structures

(Candidate to) Induced pseudo-metric

Given an ε-hypersymplectic structure, we define a vector bundle morphism g : A → A∗ by setting g := ε3ε2 ω3♭ ◦ π1♯ ◦ ω2♭ = ε1ε3 ω1♭ ◦ π2♯ ◦ ω3♭ Notice that the definition of g is not affected by a circular permutation of the indices.

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 12 / 35

Hypersymplectic structures on Lie algebroids Hypersymplectic structures

(Candidate to) Induced pseudo-metric

Given 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 of the indices.

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 12 / 35

Hypersymplectic structures on Lie algebroids Hypersymplectic structures

(Candidate to) Induced pseudo-metric

Given 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 of the indices. Then, we set as definition of g, for any i ∈ {1, 2, 3}, g := εi−1εi+1 ωi−1♭ ◦ πi ♯ ◦ ωi+1♭.

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 12 / 35

Hypersymplectic structures on Lie algebroids Hypersymplectic structures

Induced pseudo-metric

Consider (ω1, ω2, ω3) a (para-)hypersymplectic structure, i.e., such that ε1ε2ε3 = −1. Then the morphism g : Γ(A) → Γ(A∗) is a pseudo-metric on A → M, i.e., g is    C ∞(M)-linear; symmetric; non-degenerate.

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 13 / 35

Hypersymplectic structures on Lie algebroids Hypersymplectic structures

Induced pseudo-metric

Consider (ω1, ω2, ω3) a (para-)hypersymplectic structure, i.e., such that ε1ε2ε3 = −1. Then the morphism g : Γ(A) → Γ(A∗) is a pseudo-metric on A → M, i.e., g is    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).

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 13 / 35

Hypersymplectic structures on Lie algebroids Hypersymplectic structures

Induced pseudo-metric

Consider (ω1, ω2, ω3) a (para-)hypersymplectic structure, i.e., such that ε1ε2ε3 = −1. Then the morphism g : Γ(A) → Γ(A∗) is a pseudo-metric on A → M, i.e., g is    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.

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 13 / 35

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

Relation with (para-)hyperkähler

DEFINITION

A 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).

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 14 / 35

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

Relation with (para-)hyperkähler

DEFINITION

A 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).

DEFINITION

A quadruple (I1, I2, I3, g) is a para-hyperkähler structure if the following is satisfied        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.

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 14 / 35

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

Relation with (para-)hyperkähler

THEOREM

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

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 15 / 35

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

Induced structures and vice-versa

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

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 16 / 35

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

Induced structures and vice-versa

We defined the morphisms g, Ij, j = 1, 2, 3, starting from an ε-hypersymplectic structure (ω1, ω2, ω3). But the process can be reversed.

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 16 / 35

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

Induced structures and vice-versa

We defined the morphisms g, Ij, j = 1, 2, 3, starting from an ε-hypersymplectic structure (ω1, ω2, ω3). But the process can be reversed. In fact, we have ωj ♭ = ǫjǫj−1 g ◦ Ij

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 16 / 35

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

1-1 correspondence

THEOREM

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

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 17 / 35

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 in R4 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.

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 18 / 35

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 in R4 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 of sections Γ(∧2(T ∗R4)).

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 18 / 35

Hypersymplectic structures on Lie algebroids Examples in T(R4)

Examples in T(R4)

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

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 19 / 35

Hypersymplectic structures on Lie algebroids Examples in T(R4)

Examples in T(R4)

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

1 The triples (ω1, ω2, ω3) and

(ω4, ω5, ω6) are hypersymplectic structures.

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 19 / 35

Hypersymplectic structures on Lie algebroids Examples in T(R4)

Examples in T(R4)

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

1 The triples (ω1, ω2, ω3) and

(ω4, ω5, ω6) are hypersymplectic structures.

2 The 9 triples (ωi, ωj, ωk) where

1 ≤ i ≤ j ≤ 3 and k ∈ {4, 5, 6} are para-hypersymplectic structures.

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 19 / 35

Hypersymplectic structures on Lie algebroids Examples in T(R4)

Examples in T(R4)

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

1 The triples (ω1, ω2, ω3) and

(ω4, ω5, ω6) are hypersymplectic structures.

2 The 9 triples (ωi, ωj, ωk) where

1 ≤ i ≤ j ≤ 3 and k ∈ {4, 5, 6} are para-hypersymplectic structures.

3 The 9 triples (ωi, ωj, ωk) where

4 ≤ i ≤ j ≤ 6 and k ∈ {1, 2, 3} are para-hypersymplectic structures.

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 19 / 35

Hypersymplectic structures on Courant algebroids

Outline

1

Hypersymplectic structures on Lie algebroids Basics on Lie algebroids Symplectic structures Hypersymplectic structures Relation with (para-)hyperkähler structures Examples in T(R4)

2

Hypersymplectic structures on Courant algebroids Definition and examples Properties Relation with (para-)hyperkähler structures

3

Some references

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 20 / 35

Hypersymplectic structures on Courant algebroids Definition and examples

Courant algebroids

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

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 21 / 35

Hypersymplectic structures on Courant algebroids Definition and examples

Courant algebroids

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

ρ

• TM
• M

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 21 / 35

Hypersymplectic structures on Courant algebroids Definition and examples

Courant algebroids

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

ρ

• TM
• M

·, · symmetric non-degenerate bilinear form on E

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 21 / 35

Hypersymplectic structures on Courant algebroids Definition and examples

Courant algebroids

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

ρ

• TM
• M

·, · 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]]

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 21 / 35

Hypersymplectic structures on Courant algebroids Definition and examples

Courant algebroids

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

ρ

• TM
• M

·, · 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]

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 21 / 35

Hypersymplectic structures on Courant algebroids Definition and examples

Courant algebroids

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

ρ

• TM
• M

·, · 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 ]

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 21 / 35

Hypersymplectic structures on Courant algebroids Definition and examples

Examples of Courant algebroids

Generalized tangent bundle TM ⊕ T ∗M TM ⊕ T ∗M

IdTM ⊕ 0

• TM
• M

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 22 / 35

Hypersymplectic structures on Courant algebroids Definition and examples

Examples of Courant algebroids

Generalized tangent bundle TM ⊕ T ∗M TM ⊕ T ∗M

IdTM ⊕ 0

• TM
• M

◮ X + α, Y + β = iXβ + iY α

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 22 / 35

Hypersymplectic structures on Courant algebroids Definition and examples

Examples of Courant algebroids

Generalized tangent bundle TM ⊕ T ∗M TM ⊕ T ∗M

IdTM ⊕ 0

• TM
• M

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

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 22 / 35

Hypersymplectic structures on Courant algebroids Definition and examples

Examples of Courant algebroids

Generalized tangent bundle TM ⊕ T ∗M TM ⊕ T ∗M

IdTM ⊕ 0

• TM
• M

◮ X + α, Y + β = iXβ + iY α ◮ [X + α, Y + β] = [X, Y ] + (LXβ − iY dα) Starting from a Lie algebroid structure on A, we define analogously a Courant algebroid structure on A ⊕ A∗.

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 22 / 35

Hypersymplectic structures on Courant algebroids Definition and examples

Generalized terminology

A generalized structure is a structure defined on a Courant algebroid of split type, i.e., E = A ⊕ A∗, in a similar way the equivalent structure is defined on the Lie algebroid on A.

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 23 / 35

Hypersymplectic structures on Courant algebroids Definition and examples

Generalized terminology

A generalized structure is a structure defined on a Courant algebroid of split type, i.e., E = A ⊕ A∗, in a similar way the equivalent structure is defined on the Lie algebroid on A. For example, a generalized complex structure on A is a vector bundle morphism I : A ⊕ A∗ → A ⊕ A∗ satisfying I 2 = −IdA⊕A∗ and TI = 0.

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 23 / 35

Hypersymplectic structures on Courant algebroids Definition and examples

Generalized terminology

A generalized structure is a structure defined on a Courant algebroid of split type, i.e., E = A ⊕ A∗, in a similar way the equivalent structure is defined on the Lie algebroid on A. For example, a generalized complex structure on A is a vector bundle morphism I : A ⊕ A∗ → A ⊕ A∗ satisfying I 2 = −IdA⊕A∗ and TI = 0.

Remark

On a Courant algebroid E, the Nijenhuis torsion TN of a morphism N : E → E is defined as in the Lie algebroid case:

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

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 23 / 35

Hypersymplectic structures on Courant algebroids Definition and examples

Definition

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

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 24 / 35

Hypersymplectic structures on Courant algebroids Definition and examples

Definition

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

DEFINITION

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

1 Si 2 = εiIdE, 2 TSi = 0,

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

3 SiSj = ε1ε2ε3 SjSi. Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 24 / 35

Hypersymplectic structures on Courant algebroids Definition and examples

Definition

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

DEFINITION

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

1 Si 2 = εiIdE, 2 TSi = 0,

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

3 SiSj = ε1ε2ε3 SjSi.

Note

When E is a Courant algebroid of split type, i.e., E = A ⊕ A∗, (S1, S2, S3) is said a generalized ε-hypersymplectic structure on A.

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 24 / 35

Hypersymplectic structures on Courant algebroids Definition and examples

Example on A ⊕ A∗

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

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 25 / 35

Hypersymplectic structures on Courant algebroids Definition and examples

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 endomorphism Si : A ⊕ A∗ → A ⊕ A∗ given in a matrix form by Si := εi πi ωi

• .

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 25 / 35

Hypersymplectic structures on Courant algebroids Definition and examples

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 endomorphism Si : A ⊕ A∗ → A ⊕ A∗ given in a matrix form by Si := εi πi ωi

• .

The morphisms Si satisfy the following ◮ Si 2 = εi IdA⊕A∗ ◮ TSi = 0 ◮ SiSj = ε1ε2ε3 SjSi

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 25 / 35

Hypersymplectic structures on Courant algebroids Definition and examples

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 endomorphism Si : A ⊕ A∗ → A ⊕ A∗ given in a matrix form by Si := εi πi ωi

• .

The morphisms Si satisfy the following ◮ Si 2 = εi IdA⊕A∗ ◮ TSi = 0 ◮ SiSj = ε1ε2ε3 SjSi The triple (S1, S2, S3) is an ε-hypersymplectic structure on the Courant algebroid A ⊕ A∗

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 25 / 35

Hypersymplectic structures on Courant algebroids Definition and examples

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 endomorphism Si : A ⊕ A∗ → A ⊕ A∗ given in a matrix form by Si := εi πi ωi

• .

The morphisms Si satisfy the following ◮ Si 2 = εi IdA⊕A∗ ◮ TSi = 0 ◮ SiSj = ε1ε2ε3 SjSi The triple (S1, S2, S3) is an ε-hypersymplectic structure on the Courant algebroid A ⊕ A∗ or, more precisely, a generalized ε-hypersymplectic structure on A.

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 25 / 35

Hypersymplectic structures on Courant algebroids Definition and examples

Induced transition tensors and metric

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

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 26 / 35

Hypersymplectic structures on Courant algebroids Definition and examples

Induced transition tensors and metric

Consider (S1, S2, S3) an ε-hypersymplectic structure on the Courant algebroid (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.

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 26 / 35

Hypersymplectic structures on Courant algebroids Definition and examples

Induced transition tensors and metric

Consider (S1, S2, S3) an ε-hypersymplectic structure on the Courant algebroid (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.

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 26 / 35

Hypersymplectic structures on Courant algebroids Definition and examples

Continued example on A ⊕ A∗

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

• .

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 27 / 35

Hypersymplectic structures on Courant algebroids Definition and examples

Continued example on A ⊕ A∗

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

• .

◮ The respective transition tensors are given, in matrix form, by Ti = Ii ε1ε2ε3 Ii ∗

• Paulo Antunes (Univ. Coimbra)

Generalized hypersymplectic Olhão, 06/09/12 27 / 35

Hypersymplectic structures on Courant algebroids Definition and examples

Continued example on A ⊕ A∗

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

• .

◮ The respective transition tensors are given, in matrix form, by Ti = Ii ε1ε2ε3 Ii ∗

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

G := g−1 g

• Paulo Antunes (Univ. Coimbra)

Generalized hypersymplectic Olhão, 06/09/12 27 / 35

Hypersymplectic structures on Courant algebroids Properties

Properties

Consider (S1, S2, S3) an ε-hypersymplectic structure on the Courant algebroid (E, ρ, ·, ·, [·, ·]). The transition tensors satisfy ◮ Ti 2 = εiIdE; ◮ TiTj = ε1ε2ε3TjTi; ◮ Ti ∗ = ε1ε2ε3Ti; ◮ T3T2T1 = IdE.

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 28 / 35

Hypersymplectic structures on Courant algebroids Properties

Properties

Consider (S1, S2, S3) an ε-hypersymplectic structure on the Courant algebroid (E, ρ, ·, ·, [·, ·]). The transition tensors satisfy ◮ Ti 2 = εiIdE; ◮ TiTj = ε1ε2ε3TjTi; ◮ Ti ∗ = ε1ε2ε3Ti; ◮ T3T2T1 = IdE. ◮ As in Lie algebroids, in order to have objects with nice properties to work with (skew-symmetric, anti-commuting, vanishing Nijenhuis torsion. . . ) we will restrict our study to the case where ε1ε2ε3 = −1. ◮ Then, we have to distinguish two cases: ε1 = ε2 = ε3 = −1 → hypersymplectic structure on E;

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 28 / 35

Hypersymplectic structures on Courant algebroids Properties

Properties

Consider (S1, S2, S3) an ε-hypersymplectic structure on the Courant algebroid (E, ρ, ·, ·, [·, ·]). The transition tensors satisfy ◮ Ti 2 = εiIdE; ◮ TiTj = ε1ε2ε3TjTi; ◮ Ti ∗ = ε1ε2ε3Ti; ◮ T3T2T1 = IdE. ◮ As in Lie algebroids, in order to have objects with nice properties to work with (skew-symmetric, anti-commuting, vanishing Nijenhuis torsion. . . ) we will restrict our study to the case where ε1ε2ε3 = −1. ◮ 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.

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 28 / 35

Hypersymplectic structures on Courant algebroids Properties

Properties

Consider (S1, S2, S3) an ε-hypersymplectic structure on the Courant algebroid (E, ρ, ·, ·, [·, ·]). The transition tensors satisfy ◮ Ti 2 = εiIdE; ◮ TiTj = ε1ε2ε3TjTi; ◮ Ti ∗ = ε1ε2ε3Ti; ◮ T3T2T1 = IdE. ◮ As in Lie algebroids, in order to have objects with nice properties to work with (skew-symmetric, anti-commuting, vanishing Nijenhuis torsion. . . ) we will restrict our study to the case where ε1ε2ε3 = −1. ◮ 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.

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 28 / 35

Hypersymplectic structures on Courant algebroids Properties

Transition tensors are integrable

PROPOSITION

If (S1, S2, S3) is a (para-)hypersymplectic structure, then Ti is a complex tensor, if εi = −1; para-complex tensor, if εi = 1.

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 29 / 35

Hypersymplectic structures on Courant algebroids Properties

Transition tensors are integrable

PROPOSITION

If (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 Nijenhuis tensor on E.

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 29 / 35

Hypersymplectic structures on Courant algebroids Properties

G is in fact a metric

DEFINITION

A metric on (E, ρ, ·, ·, [·, ·]) is an orthogonal and symmetric bundle automorphism G : E → E.

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 30 / 35

Hypersymplectic structures on Courant algebroids Properties

G is in fact a metric

DEFINITION

A metric on (E, ρ, ·, ·, [·, ·]) is an orthogonal and symmetric bundle automorphism G : E → E.

PROPOSITION

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

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 30 / 35

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

Relation with (para-)hyperkähler structures

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

DEFINITION

A pair (G, T ) is a para-hermitian structure if the following is satisfied    G is a metric T is a para-complex tensor G(T X), T Y = −G(X), Y , for all X, Y ∈ Γ(E).

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 31 / 35

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

Relation with (para-)hyperkähler structures

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

DEFINITION

A pair (G, T ) is a para-hermitian structure if the following is satisfied    G is a metric T is a para-complex tensor G(T X), T Y = −G(X), Y , for all X, Y ∈ Γ(E).

PROPOSITION

When (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 an hermitian structure).

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 31 / 35

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

Relation with para-hyperkähler

DEFINITION

A quadruple (T1, T2, T3, G) is a para-hyperkähler structure if the following is satisfied        G is a metric T1, T2 are anti-commuting para-complex tensors and T3 = T1T2 (G, Tj)j=1,2 are para-hermitian structures

T(GTj) = 0, j = 1, 2, 3.

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 32 / 35

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

Relation with para-hyperkähler

DEFINITION

A quadruple (T1, T2, T3, G) is a para-hyperkähler structure if the following is satisfied        G is a metric T1, T2 are anti-commuting para-complex tensors and T3 = T1T2 (G, Tj)j=1,2 are para-hermitian structures

T(GTj) = 0, j = 1, 2, 3.

THEOREM

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

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 32 / 35

Some references

Outline

1

Hypersymplectic structures on Lie algebroids Basics on Lie algebroids Symplectic structures Hypersymplectic structures Relation with (para-)hyperkähler structures Examples in T(R4)

2

Hypersymplectic structures on Courant algebroids Definition and examples Properties Relation with (para-)hyperkähler structures

3

Some references

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 33 / 35

Some references

Some references

• P. Xu, Hyper-Lie Poisson structures, Annales Scientifiques de l’École

Normale Supérieure 30, Issue 3, 279–302 (1997).

• P. Antunes, Crochets de Poisson gradués et applications: structures

compatibles et généralisations des structures hyperkählériennes, Thèse de doctorat de l’École Polytechnique, (2010).

• H. Bursztyn, G. Cavalcanti and M. Gualtieri, Generalized Kähler and

hyper-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 di Torino, Classe de Scienze fisiche Matematiche e Naturali, Suplemento al numero 124, 1990, 169–180.

• P. Antunes, C. Laurent-Gengoux, J. M. Nunes da Costa, Hierarchies

and compatibility on Courant algebroids, arXiv:1111.0800.

Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 34 / 35

Some references

THE END

Thank you for your attention

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