On the structure of split regular Hom-Lie–Rinehart algebras

Abstract

The aim of this paper is to study the structures of split regular Hom-Lie Rinehart algebras. Let (L, A) be a split regular Hom-Lie Rinehart algebra. We first show that L is of the form L = U + [γ]∈Γ/∼ I [γ] with U a vector space complement in H and I [γ] are well described ideals of L satisfying I [γ] , I [δ] = 0 if I [γ] = I [δ] . Also, we discuss the weight spaces and decompositions of A and present the relation between the decompositions of L and A. Finally, we consider the structures of tight split regular Hom-Lie Rinehart algebras.There is a growing interest in twisted algebraic structures or Hom-algebraic structures defined for classical algebras and Lie algebroids as well. Hom-algebras were first introduced in the Lie algebra setting [16] with motivation from physics though its origin can be traced back in earlier literature such as [18]. In [23], Makhlouf and Silvestrov introduced the definition of Hom-associative algebras, where the associativity of a Hom-algebra is twisted by an endomorphism. Recently, Mandal and Mishra introduced the notion of Hom-Lie Rinehart algebras in [24], which is a generalization of Lie-Rinehart algebras.The class of the split algebras is specially related to addition quantum numbers, graded contractions and deformations. For instance, for a physical system which displays a symmetry of L, it is interesting to know in detail the structure of the split decomposition because its roots can be seen as certain eigenvalues which are the additive quantum numbers characterizing the state of such system. Determining the structure of split algebras will become more and more meaningful in the area of research in mathematical physics. Recently,in [1,2,4,5,6,7,8,9,10,11,12], the structure of different classes of split algebras have been determined by the techniques of connections of roots.In the present paper we introduce the class of split regular Hom-Lie Rinehart algebras as the natural extension of the one of split Lie-Rinehart algebras and so of split regular Hom-Lie algebras, and study its tight structures based on some work in [1] and [3]. In section 2, we establish the preliminaries on split regular Hom-Lie Rinehart algebras theory. In sections 3 and 4, we develop techniques of connections of roots and weights for split Hom-Lie Rinehart algebras respectively. In section 5, we study the structures of tight split regular Hom-Lie Rinehart algebras. PreliminariesIn this section, we start by recalling the definition of Hom-Lie Rinehart algebras, then we introduce the notions of roots and weights of split Hom-Lie Rinehart algebras. Throughout the paper, all algebraic systems are supposed to be over a field k. Definition 2.1. ([25]) Let A be an associative commutative algebra and φ : A → A an algebra endomorphism. A φ-derivation on A is a linear map D : A → A satisfying D(ab) = φ(a)D(b) + D(a)φ(b), (2. 1) for all a, b ∈ A. The set of all φ-derivations on A is denoted by Der φ (A).Remark 2.2. The triple (Der φ (A), [·, ·] φ , ψ φ ) is a Hom-Lie algebra, where the bracket [·, ·] φ and the ...

On split regular Hom-Leibniz-Rinehart algebras

“…Let us introduce the notion of root-multiplicativity in the framework of split regular Hom-Leibniz-Rinehart algebras of maximal length, in a similar way to the ones for split regular Hom-Lie Rinehart algebras in [26].…”

“…Recently, in [1]- [7], the structure of different classes of split algebras have been determined by the techniques of connections of roots. Recently, we studied the structures of split regular Hom-Lie Rinehart algebras in [26]. The purpose of this paper is to consider the structure of split regular Hom-Leibniz-Rinehart algebras by the techniques of connections of roots based on some work in [7] and [26] .…”

“…Recently, we studied the structures of split regular Hom-Lie Rinehart algebras in [26]. The purpose of this paper is to consider the structure of split regular Hom-Leibniz-Rinehart algebras by the techniques of connections of roots based on some work in [7] and [26] .…”

Hom 3-Lie-Rinehart Algebras

“…For example, it is applied to the study of Nambu mechanics and the study of supersymmetry and gauge symmetry transformations of the world-volume theory of multiple coincident M2-branes [1,20,21,23]. The authors of [3,4,10,22,24] studied the symplectic structures on 3-Lie algebras, 3-Lie bialgebras and 3-Lie Yang-Baxter equation, and constructed the tensor form of skew-symmetric solutions of 3-Lie Yang-Baxter equation in 3-Lie algebras. In [12,13,14], Casas introduced the cross modules of Lie-Rinehart algebras and proved that three cohomology and Rinehart cohomology are isomorphic if Lie-Rinehart algebras are projected onto the corresponding commutative algebras.…”

On 3-Hom-Lie-Rinehart algebras

“…The associated deformation cohomology that controls deformations was constructed using multiderivations of Hom-Lie-Rinehart algebras in [15]. Moreover, Zhang et al stuied crossed modules for Hom-Lie-Rinehart algebras in [21], we studied the structures of split regular Hom-Lie Rinehart algebras in [20].…”