Nijenhuis Operators On Left-symmetric Algebras(Algebroids)

Nijenhuis operators were originally introduced by A.Nijenhuis in 1951 to characterize the integrability of a distribution associated to the eigenvectors of a special tensor of type(1,1).People find that Nijenhuis operators have a close connection with many fields in mathematics and mathematical physics such as deformation of algebras,integrability condition in differential geometry,integrable system and so on.In this thesis,we mainly study the properties of Nijenhuis operators and their applications in left-symmetric algebras and left-symmetric algebroids.For the Nijenhuis operators on left-symmetric algebras,we study the relationships between Nijenhuis operators and O-operators,Rota-Baxter operators,s-matrices,pseudo-Hessian structures and para-complex structures on left-symmetric algebras.For the Nijenhuis structures on left-symmetric algebras with representations,we introduce the notion of an ON-structure on a left-symmetric algebra with a representation and use the strong Maurer-Cartan elements on a twilled left-symmetric algebras to construct ON-structures.For the Nijenhuis operators on left-symmetric algebroids,we introduce the notion of a Koszul-Vinberg-Nijenhuis structure and study the relations between this structure and KVΩ-structures,pseudo-Hessian-Nijenhuis structures and complementary symmetric 2-tensors for Koszul-Vinberg structures on left-symmetric algebroids.For a vector space g,we first define a graded Lie algebra on the complex ⊕kHom(∧kg(?)g,g)whose Maurer-Cartan elements characterize left-symmetric algebra structures.Then using this graded Lie bracket we define the notion of a Nijenhuis operator on a left-symmetric algebra which generates a trivial deformation of this left-symmetric algebra.There are close relationships between O-operators,Rota-Baxter operators and Nijenhuis operators on a left-symmetric algebra.In particular,for two invertible O-operators on a left-symmetric algebra whose any linear combination is still an O-operator if and only if N:=T1(?)T2-1 is a Nijenhuis operator.L-dendriform algebras are the underlying algebra structures of an O-operator and hence compatible L-dendriform algebras appear naturally as the induced algebraic structures.For the case of the dual representation of the regular representation of an O-operator,there is a geometric interpretation by introducing the notion of a pseudo-Hessian-Nijenhuis structure which gives rise to a sequence of pseudo-Hessian and pseudo-Hessian-Nijenhuis structures.We study the relationship between compatible 5-matrices and pseudo-Hessian-Nijenhuis structures.Another application of Nijenhuis operators on left-symmetric algebras in geometry is illustrated by introducing the notion of a para-complex structure on a left-symmetric algebra and then studying para-complex quadratic left-symmetric algebras and para-complex pseudo-Hessian left-symmetric algebras in detail.We give some examples of Nijenhuis operators on left-symmetric algebras from various aspects.We study infinitesimal deformations on left-symmetric algebras with a representation and introduce the notion of a Nijenhuis structure on a left-symmetric algebra with a representation.Then we add compatibility conditions between an O-operator and a Nijenhuis structure to define the notion of an ON-structure on a left-symmetric algebra with a representation.We use compatible O-operators and strong Maurer-Cartan elements on the twilled left-symmetric algebras associated to an O-operator to construct ON-structures.As a special case of ON-structures,we give the notion of s-matrix-Nijenhuis structures.Koszul-Vinberg structures are introduced by Yunhe Sheng and his coauthors in their study of left-symmetric bialgebroids,which are geometric generalization of 5-matrices on left-symmetric algebras.Similarly to a Poisson structure on a Lie algebroid giving a Lie bialgebroid,a Koszul-Vinberg structure on a left-symmetric algebroid gives a left-symmetric bialgebroid.We introduce the notion of Koszul-Vinberg-Nijenhuis structures on a left-symmetric algebroid as analogues of Poisson-Nijenhuis structures on a Lie algebroid,and show that a Koszul-Vinberg-Nijenhuis structure gives rise to a hierarchy of compatible Koszul-Vinberg structures.We introduce the notions of KVΩ-structures,pseudo-Hessian-Nijenhuis structures and complementary symmetric 2-tensors for Koszul-Vinberg structures on left-symmetric algebroids,which are analogues of PΩ-structures,symplectic-Nijenhuis structures and complementary 2-forms for Poisson structures.We also study the relationships between these various structures.