The purpose of this paper is to study hom-algebroids, among them left symmetric homalgebroids and symplectic hom-algebroids by providing some characterizations and geometric interpretations. Therefore, we introduce and study para-Kähler hom-Lie algebroids and show various properties and examples including these structures.2010 Mathematics Subject Classification. 17A30, 17D25, 53C15, 53D05. Key words and phrases.Hom-algebroid, left symmetric hom-algebroid, Hom-Lie algebroid, para-Kähler hom-Lie algebroid.
Complex and Hermitian structures on hom-Lie algebras are introduced and some examples of these structures are presented. Also, it is shown that there not exists a proper complex (Hermitian) home-Lie algebra of dimension two. Then using a hom-left symmetric algebra, a phase space is provided and then a complex structure on it, is presented. Finally, the notion of Kähler hom-Lie algebra is introduced and then using a Kähler hom-Lie algebra, a phase space is constructed. IntroductionStudying the deformations of the Witt and Virasoro algebras, lead Hartwig, Larsson and Silvestrov to introduce the notion of hom-Lie algebras [5]. Indeed, some q-deformations of the Witt and the Virasoro algebras have the structure of a hom-Lie algebra [5,6]. Based on the close relation between the discrete, deformed vector fields and differential calculus, this algebraic structure plays important role in research fields [5,7].Differential-geometric structures play important role in the study of complex geometry. After Kodaira, Kähler structures became central in the study of deformation theory and the classification problems.Recall that an almost complex structure J on a manifold M is a linear complex structure (that is, a linear map which squares to -1) on each tangent space of the manifold, which varies smoothly on the manifold. A complex structure is essentially an almost complex structure with an integrability condition, and this condition yields an unitary structure (U (n) structure) on the manifold. An almost Hermitian structure is a pair (J, g) of an almost complex structure J and a pseudo-Riemannian metric g such that g(·, ·) = g(J·, J·). A manifold M is called almost Hermitian manifold if it is endowed with an almost Hermitian structure (J, g). An almost Hermitian manifold (M, J, g) is called Kähler, if its Levi-Civita connection ∇ satisfies ∇J = 0.Recently, some mathematicians have studied some geometric concepts over Lie groups and Lie algebras such as complex, complex product and contact structures [1,2,4]. Inspired by these papers, the same authors in [8] introduced some geometric concepts such as para-complex, para-Hermitian and para-Kähler structures on hom-Lie algebras. The aim of this paper is introducing other geometric concepts on these algebras.The structure of this paper is organized as follows: In Section 2, we recall the definition of hom-algebra, hom-left-symmetric algebra, hom-Lie algebra, symplectic hom-Lie algebra and pseudo-Riemannian homalgebra and hom-Levi-Civita product. Also we present an example of a symplectic hom-Lie algebra.In Section 3, we introduce complex and Hermitian structures on hom-Lie algebras and we determine all complex hom-Lie algebra of dimension 2. Also, we show that there not exists a proper complex (Hermitian) hom-Lie algebra of dimension 2. Moreover, we present an example of an almost Hermitian 2010 Mathematics Subject Classification. 17A30, 17D25, 53C15, 53D05.Key words and phrases. almost Hermitian structure, hom-Levi-Civita product, Kähler hom-Lie algebra, phase space. ESMAEIL PEYG...
In this paper, we describe the construction of connections on hom-bundles and the pseudo-Riemannian structure on hom-Lie algebroids from an algebraic point of view. We study the representations of hom-Lie algebroids. We introduce the notion of a hom-left symmetric algebroid as a geometric generalization of a hom-left symmetric algebra. In addition using [Formula: see text]-operators, we construct some classes of hom-left symmetric algebroids. We show that there exists a phase space on a hom-left symmetric algebroid. Also, we prove that using phase spaces, we can construct hom-left symmetric algebroids.
In this paper, we construct K?hler-Norden statistical structures on pseudo-Riemannian manifolds equipped with a torsion-free linear connection and an almost complex structure. Also, we present some examples and study curvature properties for these structures by using Tachibana operator. Finally, we consider a Norden statistical manifold and study Codazzi coupling of its connection with the almost complex structure on it.
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