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 Hypersymplectic structures on Lie algebroids 1 Basics on Lie algebroids Symplectic structures Hypersymplectic structures Relation with (para-)hyperkähler structures Examples in T ( R 4 ) Hypersymplectic structures on Courant algebroids 2 Definition and examples Properties Relation with (para-)hyperkähler structures Some references 3 Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 2 / 35
Hypersymplectic structures on Lie algebroids Outline Hypersymplectic structures on Lie algebroids 1 Basics on Lie algebroids Symplectic structures Hypersymplectic structures Relation with (para-)hyperkähler structures Examples in T ( R 4 ) Hypersymplectic structures on Courant algebroids 2 Definition and examples Properties Relation with (para-)hyperkähler structures Some references 3 Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 3 / 35
� � Hypersymplectic structures on Lie algebroids Basics on Lie algebroids Definition ρ A TM τ Consider M a smooth manifold and A − → M � � � � τ � � � a vector bundle. � � � � � � � � � � M τ A Lie algebroid structure on A → 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 id TM TM � � � � � τ τ 1) Tangent bundle TM � � � � � � � � � � � � M Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 5 / 35
� � � � Hypersymplectic structures on Lie algebroids Basics on Lie algebroids Examples id TM TM � � � � � τ τ 1) Tangent bundle TM � � � � � � � � � � � � M ρ ≡ 0 { 0 } g � � � � � � 2) Lie algebra g � � � � � � � � � � � {∗} Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 5 / 35
� � � � � � Hypersymplectic structures on Lie algebroids Basics on Lie algebroids Examples id TM TM � � � � � τ τ 1) Tangent bundle TM � � � � � � � � � � � � M ρ ≡ 0 { 0 } g � � � � � � 2) Lie algebra g � � � � � � � � � � � {∗} 3) Cotangent bundle T ∗ M of a Poisson manifold π ♯ ( M , π ) , equipped with the bracket T ∗ M TM � � � � � � � � � [ α, β ] π = L π # ( α ) β − L π # ( β ) α − d ( π ( α, β )) , � � � � � � � � M for all α, β ∈ Γ( T ∗ 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 or, equivalently, �� � T I ( X , Y ) = 1 � [ X , Y ] I I − [ X , Y ] I 2 . 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 or, equivalently, �� � T I ( X , Y ) = 1 � [ X , Y ] I I − [ X , Y ] I 2 . 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 = − Id A ; 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 = − Id A ; T I = 0 . The ( 1 , 1 ) -tensor I ∈ Γ( A ⊗ A ∗ ) is a para-complex tensor if it satisfies � I 2 = Id A ; 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 ∗ )) , k � ( − 1 ) i ρ ( X i ) · η ( X 0 , . . . , � d η ( X 0 , . . . , X k ) := X i , . . . , X k ) i = 0 �� � � � ( − 1 ) i + j η , X 0 , . . . , � X i , . . . , � + X i , X j X j , . . . , X k , 0 ≤ i < j ≤ k for all X 0 , . . . , X k ∈ 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 ∗ )) , k � ( − 1 ) i ρ ( X i ) · η ( X 0 , . . . , � d η ( X 0 , . . . , X k ) := X i , . . . , X k ) i = 0 �� � � � ( − 1 ) i + j η , X 0 , . . . , � X i , . . . , � + X i , X j X j , . . . , X k , 0 ≤ i < j ≤ k for all X 0 , . . . , X k ∈ Γ( A ) , where the symbol � means that the term below is omitted. Examples If A = TM (and ρ = Id ), then d is the de Rham differential. 1 If A = g is a Lie algebra, then d is the Chevalley-Eilenberg differential. 2 If A = T ∗ M, with ( M , π ) a Poisson manifold, then d is the Lichnerowicz 3 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( ∧ 2 A ∗ ) which is non-degenerate, i.e., ∃ π ∈ Γ( ∧ 2 A ) such that π ♯ ◦ ω ♭ = Id A ; closed, i.e, such that d ω = 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 ω 1 , ω 2 and ω 3 ∈ Γ( ∧ 2 A ∗ ) with inverse Poisson bivec- tors π 1 , π 2 and π 3 ∈ Γ( ∧ 2 A ) , respectively. We define the transition ( 1 , 1 ) -tensors I 1 , I 2 , I 3 : Γ( A ) → Γ( A ) , by setting I j := π ♯ j − 1 ◦ ω ♭ j + 1 . considering the indices as elements of Z 3 . Paulo Antunes (Univ. Coimbra) Generalized hypersymplectic Olhão, 06/09/12 10 / 35
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