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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
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 2 / 33
Hypersymplectic structures on Lie algebroids
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
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 3 / 33
Hypersymplectic structures on Lie algebroids Basics on Lie algebroids
Definition
Consider M a smooth manifold and A τ−→ Ma vector bundle.
Aρ //
τ
��@@@
@@@@
@ TM
}}zzzz
zzzz
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).
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 4 / 33
Hypersymplectic structures on Lie algebroids Basics on Lie algebroids
Examples
1) Tangent bundle TMTM
id //
τ
!!CCC
CCCC
C TMτ
}}{{{{
{{{{
M
2) Lie algebra g
gρ≡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 ∗Mπ] //
""EEE
EEEE
E TM
}}{{{{
{{{{
M
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 5 / 33
Hypersymplectic structures on Lie algebroids Basics on Lie algebroids
Examples
1) Tangent bundle TMTM
id //
τ
!!CCC
CCCC
C TMτ
}}{{{{
{{{{
M
2) Lie algebra g
gρ≡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 ∗Mπ] //
""EEE
EEEE
E TM
}}{{{{
{{{{
M
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 5 / 33
Hypersymplectic structures on Lie algebroids Basics on Lie algebroids
Examples
1) Tangent bundle TMTM
id //
τ
!!CCC
CCCC
C TMτ
}}{{{{
{{{{
M
2) Lie algebra g
gρ≡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 ∗Mπ] //
""EEE
EEEE
E TM
}}{{{{
{{{{
M
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 5 / 33
Hypersymplectic structures on Lie algebroids Basics on Lie algebroids
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.
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 6 / 33
Hypersymplectic structures on Lie algebroids Basics on Lie algebroids
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.
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 6 / 33
Hypersymplectic structures on Lie algebroids Basics on Lie algebroids
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.
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 6 / 33
Hypersymplectic structures on Lie algebroids Basics on Lie algebroids
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.
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 6 / 33
Hypersymplectic structures on Lie algebroids Basics on Lie algebroids
(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.
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 7 / 33
Hypersymplectic structures on Lie algebroids Basics on Lie algebroids
(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.
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 7 / 33
Hypersymplectic structures on Lie algebroids Basics on Lie algebroids
(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.
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 7 / 33
Hypersymplectic structures on Lie algebroids Basics on Lie algebroids
Exterior differential
DEFINITIONThe exterior differential d : Γ(∧•(A∗)) → Γ(∧•+1(A∗)) is defined by setting for allη ∈ Γ(∧k(A∗)),
dη(X0, . . . ,Xk) :=k∑
i=0
(−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, .]
SN.
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 8 / 33
Hypersymplectic structures on Lie algebroids Basics on Lie algebroids
Exterior differential
DEFINITIONThe exterior differential d : Γ(∧•(A∗)) → Γ(∧•+1(A∗)) is defined by setting for allη ∈ Γ(∧k(A∗)),
dη(X0, . . . ,Xk) :=k∑
i=0
(−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, .]
SN.
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 8 / 33
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 )
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 9 / 33
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).
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 10 / 33
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).
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 10 / 33
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 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.
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 11 / 33
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 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.
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 11 / 33
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 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.
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 11 / 33
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 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.
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 11 / 33
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 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.
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 11 / 33
Hypersymplectic structures on Lie algebroids Hypersymplectic structures
(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[.
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 12 / 33
Hypersymplectic structures on Lie algebroids Hypersymplectic structures
(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[.
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 12 / 33
Hypersymplectic structures on Lie algebroids Hypersymplectic structures
(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[.
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 12 / 33
Hypersymplectic structures on Lie algebroids Hypersymplectic structures
(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[.
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 12 / 33
Hypersymplectic structures on Lie algebroids Hypersymplectic structures
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.,
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) Courant hypersymplectic Burgos, 30/08/12 13 / 33
Hypersymplectic structures on Lie algebroids Hypersymplectic structures
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.,
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) Courant hypersymplectic Burgos, 30/08/12 13 / 33
Hypersymplectic structures on Lie algebroids Hypersymplectic structures
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.,
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) Courant hypersymplectic Burgos, 30/08/12 13 / 33
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.
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 14 / 33
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.
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 14 / 33
Hypersymplectic structures on Lie algebroids Relation with (para-)hyperkähler structures
Relation with (para-)hyperkähler
THEOREMIf (ω1, ω2, ω3) is a para-hypersymplectic structure then (I1, I2, I3, g) is apara-hyperkähler structure.
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 15 / 33
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 anε-hypersymplectic structure (ω1, ω2, ω3).
But the process can be reversed.
In fact, we have
ωj[ = εjεj−1 g ◦ Ij
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 16 / 33
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 anε-hypersymplectic structure (ω1, ω2, ω3).
But the process can be reversed.
In fact, we have
ωj[ = εjεj−1 g ◦ Ij
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 16 / 33
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 anε-hypersymplectic structure (ω1, ω2, ω3).
But the process can be reversed.
In fact, we have
ωj[ = εjεj−1 g ◦ Ij
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 16 / 33
Hypersymplectic structures on Lie algebroids Relation with (para-)hyperkähler structures
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.
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 17 / 33
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)).
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 18 / 33
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)).
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 18 / 33
Hypersymplectic structures on Lie algebroids Examples in 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.
3 The 9 triples (ωi , ωj , ωk) where4 ≤ i ≤ j ≤ 6 and k ∈ {1, 2, 3} arepara-hypersymplectic structures.
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 19 / 33
Hypersymplectic structures on Lie algebroids Examples in 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.
3 The 9 triples (ωi , ωj , ωk) where4 ≤ i ≤ j ≤ 6 and k ∈ {1, 2, 3} arepara-hypersymplectic structures.
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 19 / 33
Hypersymplectic structures on Lie algebroids Examples in 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.
3 The 9 triples (ωi , ωj , ωk) where4 ≤ i ≤ j ≤ 6 and k ∈ {1, 2, 3} arepara-hypersymplectic structures.
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 19 / 33
Hypersymplectic structures on Lie algebroids Examples in 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.
3 The 9 triples (ωi , ωj , ωk) where4 ≤ i ≤ j ≤ 6 and k ∈ {1, 2, 3} arepara-hypersymplectic structures.
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 19 / 33
Hypersymplectic structures on Courant algebroids
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
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 20 / 33
Hypersymplectic structures on Courant algebroids Definition and examples
Courant algebroids
Courant algebroid over M, (E , ρ, 〈·, ·〉, [·, ·]):
Eρ //
@@@
@@@@
TM
}}zzzz
zzzz
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) Courant hypersymplectic Burgos, 30/08/12 21 / 33
Hypersymplectic structures on Courant algebroids Definition and examples
Courant algebroids
Courant algebroid over M, (E , ρ, 〈·, ·〉, [·, ·]):
Eρ //
@@@
@@@@
TM
}}zzzz
zzzz
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) Courant hypersymplectic Burgos, 30/08/12 21 / 33
Hypersymplectic structures on Courant algebroids Definition and examples
Courant algebroids
Courant algebroid over M, (E , ρ, 〈·, ·〉, [·, ·]):
Eρ //
@@@
@@@@
TM
}}zzzz
zzzz
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) Courant hypersymplectic Burgos, 30/08/12 21 / 33
Hypersymplectic structures on Courant algebroids Definition and examples
Courant algebroids
Courant algebroid over M, (E , ρ, 〈·, ·〉, [·, ·]):
Eρ //
@@@
@@@@
TM
}}zzzz
zzzz
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) Courant hypersymplectic Burgos, 30/08/12 21 / 33
Hypersymplectic structures on Courant algebroids Definition and examples
Courant algebroids
Courant algebroid over M, (E , ρ, 〈·, ·〉, [·, ·]):
Eρ //
@@@
@@@@
TM
}}zzzz
zzzz
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) Courant hypersymplectic Burgos, 30/08/12 21 / 33
Hypersymplectic structures on Courant algebroids Definition and examples
Courant algebroids
Courant algebroid over M, (E , ρ, 〈·, ·〉, [·, ·]):
Eρ //
@@@
@@@@
TM
}}zzzz
zzzz
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) Courant hypersymplectic Burgos, 30/08/12 21 / 33
Hypersymplectic structures on Courant algebroids Definition and examples
Examples of Courant algebroids
Generalized tangent bundle TM ⊕ T ∗M
TM ⊕ T ∗MIdTM ⊕ 0 //
%%LLLLLLLLLL TM
}}{{{{
{{{{
M
I 〈X + α,Y + β〉 = iXβ + iYαI [X + α,Y + β] = [X ,Y ] + (LXβ − iY dα)
Double of a Lie (bi)algebroid A⊕ A∗
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 22 / 33
Hypersymplectic structures on Courant algebroids Definition and examples
Examples of Courant algebroids
Generalized tangent bundle TM ⊕ T ∗M
TM ⊕ T ∗MIdTM ⊕ 0 //
%%LLLLLLLLLL TM
}}{{{{
{{{{
M
I 〈X + α,Y + β〉 = iXβ + iYα
I [X + α,Y + β] = [X ,Y ] + (LXβ − iY dα)
Double of a Lie (bi)algebroid A⊕ A∗
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 22 / 33
Hypersymplectic structures on Courant algebroids Definition and examples
Examples of Courant algebroids
Generalized tangent bundle TM ⊕ T ∗M
TM ⊕ T ∗MIdTM ⊕ 0 //
%%LLLLLLLLLL TM
}}{{{{
{{{{
M
I 〈X + α,Y + β〉 = iXβ + iYαI [X + α,Y + β] = [X ,Y ] + (LXβ − iY dα)
Double of a Lie (bi)algebroid A⊕ A∗
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 22 / 33
Hypersymplectic structures on Courant algebroids Definition and examples
Examples of Courant algebroids
Generalized tangent bundle TM ⊕ T ∗M
TM ⊕ T ∗MIdTM ⊕ 0 //
%%LLLLLLLLLL TM
}}{{{{
{{{{
M
I 〈X + α,Y + β〉 = iXβ + iYαI [X + α,Y + β] = [X ,Y ] + (LXβ − iY dα)
Double of a Lie (bi)algebroid A⊕ A∗
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 22 / 33
Hypersymplectic structures on Courant algebroids Definition and examples
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 .
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 23 / 33
Hypersymplectic structures on Courant algebroids Definition and examples
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 .
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 23 / 33
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 endomorphismSi : A⊕ A∗ → A⊕ A∗ given in a matrix form by
Si :=
[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∗.
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 24 / 33
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 endomorphismSi : A⊕ A∗ → A⊕ A∗ given in a matrix form by
Si :=
[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∗.
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 24 / 33
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 endomorphismSi : A⊕ A∗ → A⊕ A∗ given in a matrix form by
Si :=
[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∗.
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 24 / 33
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 endomorphismSi : A⊕ A∗ → A⊕ A∗ given in a matrix form by
Si :=
[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∗.
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 24 / 33
Hypersymplectic structures on Courant algebroids Definition and examples
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.
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 25 / 33
Hypersymplectic structures on Courant algebroids Definition and examples
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.
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 25 / 33
Hypersymplectic structures on Courant algebroids Definition and examples
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.
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 25 / 33
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 :=
[0 εi πiωi 0
].
I The respective transition tensors are given, in matrix form, by
Ti :=
[Ii 00 ε1ε2ε3 Ii ∗
]I The (wannabe) metric is given, in matrix form, by
G :=
[0 g−1
g 0
]
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 26 / 33
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 :=
[0 εi πiωi 0
].
I The respective transition tensors are given, in matrix form, by
Ti :=
[Ii 00 ε1ε2ε3 Ii ∗
]
I The (wannabe) metric is given, in matrix form, by
G :=
[0 g−1
g 0
]
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 26 / 33
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 :=
[0 εi πiωi 0
].
I The respective transition tensors are given, in matrix form, by
Ti :=
[Ii 00 ε1ε2ε3 Ii ∗
]I The (wannabe) metric is given, in matrix form, by
G :=
[0 g−1
g 0
]
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 26 / 33
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 .
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 27 / 33
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 .
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 27 / 33
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 .
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 27 / 33
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 .
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 27 / 33
Hypersymplectic structures on Courant algebroids 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 .
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 28 / 33
Hypersymplectic structures on Courant algebroids 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 .
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 28 / 33
Hypersymplectic structures on Courant algebroids Properties
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.
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 29 / 33
Hypersymplectic structures on Courant algebroids Properties
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.
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 29 / 33
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).
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 30 / 33
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).
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 30 / 33
Hypersymplectic structures on Courant algebroids Relation with (para-)hyperkähler structures
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.
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 31 / 33
Hypersymplectic structures on Courant algebroids Relation with (para-)hyperkähler structures
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.
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 31 / 33
Some references
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
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 32 / 33
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.
Paulo Antunes (Univ. Coimbra) Courant hypersymplectic Burgos, 30/08/12 33 / 33
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