Some Study Of Left-symmetric Algebroids

In this thesis, we mainly study left-symmetric algebroids, pre-symplectic algebroids, left-symmetric bialgebroids and Manin triples for left-symmetric algebroids. We build the close relations between left-symmetric algebroids and Lie algebroids, pre-symplectic algebroids and symplectic Lie algebroids, left-symmetric bialgebroids and pre-symplectic algebroids. More importantly, we apply para-Kahler Lie algebroids and Hessian geometry to the theories of left-symmetric bialgebroids.We introduce the notion of a left-symmetric algebroid, which is a generalization of a left-symmetric algebra from a vector space to a vector bundle. We can obtain the sub-adjacent Lie algebroid Ac from a left-symmetric algebroid A by using the commutator. The left multiplication of the left-symmetric algebroid A gives rise to a representation of the sub-adjacent Lie algebroid Ac. We construct left-symmetric algebroids using O-operators. We extend the multiplication on Γ(A) to Γ(Λ·A), and obtain a graded Lie-admissible algebra, whose commutator is just a graded Lie algebra corresponding to the Schouten bracket of the sub-adjacent Lie algebroid Ac. We study phase spaces of Lie algebroids in terms of left-symmetric algebroids. We prove that there is a phase space (P, [.,.]P,aP,ω) of the sub-adjacent Lie algebroid Ac associated to a left-symmetric algebroid A naturally, where P = Ac×L* A*, for x, y ∈Γ(A);ζ,η∈Γ (A*), the Lie algebroid structure on P given by: the symplectic structure ω defined by: and there is a natural paracomplex structure P:P→P on the phase space P defined by Furthermore,if this left-symmetric algebroid is a pseudo-Riemannian left-symmetric algebroid (A, (·,·)+), where (·,·)+ is the pseudo-Riemannian metric on the left-symmetric algebroid A, then there is a natural complex structure J: P→P on the phase space defined by: where ψ: A→A* induced by pseudo-Riemannian metric (·,·)+, i.e. and {J, P} is a complex product structure on the phase space P, i.e. satising JP=-PJ. If (·,·)+ is a. Riemannian metic on left-symmetric algebroid A, then {J, ψ} is a Kahler structure on P such that Pis a Kahler Lie algebroid. We study representations of a left-symmetric algebroid in details and develop its cohomologw theory. We introduce a new cohomology, called the deformation cohomology associated to a left-symmetric algebroid, whose second cohomology group could control deformations of the left-symmetric algebroid. We also introduce the notion of a Nijenhuis operator, which could generate a trivial deformation. We build the differential calculus on left-symmetric algebroids. and define the Lie derivative and contraction. We. get fornmlas parallel to the classical calculus of the differential forms on manifolds, which are the foundation of further study of left-symmetric algebroids.We introduce the notion of a pre-symplectic algebroid and prove that if (E,(?), ρ, (·,·)_) is a pre-symplectic algebroid, then (E, [·,·]E, ρ, ω = (·,·)_) is a symplectic Lie algebroid, where Conversely, if (E, [·,·]E, ρ, ω) is a symplectic Lie algebroid, then (E, (?), ρ, (·,·)_ = ω) is a pre-symplectic algebroid, and satisfies where the multiplication, is defined by We introduce the notion of a Dirac structure in a pre-symplectic algebroid and prove that there is a one-to-one correspondence between Dirac structures of a pre-symplectic algebroid (E, *, p, (.,-)_) and Lagrangian subalgebroids of the corresponding symplectic Lie algebroid (E, [., "]E, P, w = (., .)_). We study para-complex pre-symplectic algebroids and prove that if (E,,, p, (., .)_, P) is a para-complex pre-symplectic algebroid, then (E, [.,.]E,ρ,ω = (’,’)-, P)is a para-Kahler Lie algebroid. Conversely, if (E, [.,.]E,ρ,ω,P) is a para-Kahler Lie algebroid, then (E, *, p, (.,.)_= w, P) is a para-complex pre-symplectic algebroid. Also, if (E,-(?), p, (., .)_, P) is a para-complex pre-symplectic algebroid, then (E, g) is a pseudo-Riemannian Lie algebroid, where g is the pseudo-Riemannian metric given by: and we prove that the multiplication of the pre-symplectic algebroid structure is same as the restriction to Lagrangian subalgebroids of the Levi-Civita connection in the corresponding pseudo-Riemannian Lie algebroid. We study exact pre-symplectic algebroids and prove that exact pre-symplectic algebroids can be classified by the third cohomology group of a left-symmetric algebroid.We introduce the notion of a leR-symmetric bialgebroid and prove that if (A, A, aA) is a left-symmetric algebroid, for H∈Sym2(A), satisfying S-equation, i.e.[H, H]=0, then (A*, ·H, aA*=aA o H#) is a left-symmetric algebroid, where H# : A*→ A is defined by: H#(ζ)(η) =H(ζ,η), and multiplication "H is defined by: and H# is a left-symmetric algebroid homomorphism from (A*, "H, aA.) to (A, "A, aA). Furthermore, (A, A*) is a left-symmetric bialgebroid. In partular, if (M, (?), g) is a pseudo-Hessian manifold, for H∈Sym2(TM) defined by: then (T*M, "H, H#) is a left-symmetric algebroid. We denote it by TH*M. H# is a left-symmetric algebroid homomorphism from T*HM to T▽M, where T▽M is the tangent left-symmetric algebroid given by pseudo-Hessian manifold, and (T▽M, TH*M) is a left-symmetric bialgebroid. The above result is parallel to that (TM, T*πM) is a Lie bialgebroid for any Poisson manifold (M, π). If (A, A*) is a left-symmetric bialgebroid, then (E = A (?)A*, *,ρ, (.,.)_) is a pre-symplectic algebroid and pre-symplectic algebroid structure given b3 Conversely, if (E, *, p; (.,.)_) is a pre-symplectic algebroid, let L1 and L2 be two transversal Dirac structures, i.e. E = L1 (?) L2, then (L1, L2) a left-symmetric bialgebroid, where L2 is considered as the dual bundle of L1 under the pairing (.,.)_. For all H ∈Γ(A(?) A). we denote by GH the graph of H1, i.e. GH = {H#(ζ) +ζ|(?)ζ∈A*}. We prove that GH is a Dirac structure of the above pre-symplectic algebroid (E = A (?) A*, *, ρ, (-, .)_) if and only if H E Syrn.2(A) and satisfies the following Maurer-Cartan type equation:...