Hom-left symmetric algebroids

Abstract

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.

Para-Kähler hom-Lie algebras

“…for any X, Y, Z ∈ Γ(A). This connection is called the hom-Levi-Civita connection [15], which is given by Koszul's formula…”

Linear Poisson structures and Hom-Lie algebroids

“…In this section, we present some notions and results about Hom-Lie algebras and Hom-Lie algebroids (see [1,9,10] for more details).…”

“…A Hom-Lie algebroid has its own geometric meaning and interesting examples, and it is more than a formal generalization of a Lie algebroid. Recently, many researchers have been interested in studying the algebraic and geometric concepts on Lie algebroids and Hom-Lie algebroids ( [3,5,7,8,10,11,12,13,14,15,16]).…”

K\"{a}hler-Norden structures on Hom-Lie groups and Hom-Lie algebras

“…Based on the close relation between the discrete, deformed vector fields and differential calculus, this algebraic structure plays an important role in various research fields [9,13,[15][16][17]21]. Recently, many algebraic and geometric problems related to Hom-Lie algebras were raised and studied, such as infinite-dimensional Hom-Lie algebras, the classical Hom-Yang-Baxter equation [25], para-Kähler and complex and Kähler structures on Hom-Lie algebras [21,22], complex product structures on Hom-Lie algebras and Hom-left symmetric algebroids [19,23] and Hom-Novikov algebras [28].…”