Split Lie–Rinehart algebras

Abstract

We introduce the class of split Lie–Rinehart algebras as the natural extension of the one of split Lie algebras. We show that if [Formula: see text] is a tight split Lie–Rinehart algebra over an associative and commutative algebra [Formula: see text] then [Formula: see text] and [Formula: see text] decompose as the orthogonal direct sums [Formula: see text] and [Formula: see text], where any [Formula: see text] is a nonzero ideal of [Formula: see text], any [Formula: see text] is a nonzero ideal of [Formula: see text], and both decompositions satisfy that for any [Formula: see text], there exists a unique [Formula: see text] such that [Formula: see text]. Furthermore, any [Formula: see text] is a split Lie–Rinehart algebra over [Formula: see text]. Also, under mild conditions, it is shown that the above decompositions of [Formula: see text] and [Formula: see text] are by means of the family of their, respective, simple ideals.

Split 3-Lie-Rinehart color algebras

“…Our goal in this work is to study the inner structure of arbitrary split 3−Lie-Rinehart color algebras by the developing techniques of connections of root systems and weight systems associated to a splitting Cartan subalgebra. The finding of the present paper is an improvement and extension of the works conducted in [2].…”

“…], ρ, ǫ) denotes a 3−Lie-Rinehart color algebra. We introduce the class of split algebras in the framework of 3−Lie-Rinehart color algebras as in [2]. Definition 2.14.…”

“…In this section, we begin by developing the techniques of connections of roots in the setting as [2]. Let (L, A) be a split 3−Lie-Rinehart color algebra, with root space decomposition L = H 0 ⊕( g∈G α∈Π g L g α ), and with symmetric root system Π = g∈G Π g .…”

“…It is worth mentioning that these techniques had long been introduced by Calderon, Antonio J, on split Lie algebras with symmetric root systems in [7]. Recently, in [2,3,9,10,12,13,14], the structure of arbitrary split Lie color algebras, Split regular Hom-Lie algebras, split Leibniz algebras, split Lie-Rinehart algebras, split 3−Lie algebras, split 3−Leibniz algebras, and arbitrary split Lie algebras of order 3 have been determined by the techniques of connection of roots.…”

Structure and cohomology of 3-Lie-Rinehart superalgebras

“…The study of 3-Lie algebras [29] gets a lot of attention since it has close relationships with Lie algebras, Hom-Lie algebras, commutative associative algebras, and cubic matrices [14][15][16]20]. 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 [10,46,51,52]. A notion of n-Lie Rinehart algebras was introduced recently in [18] and extensions and crossed modules were developed for such algebras.…”