Structure and cohomology of 3-Lie-Rinehart superalgebras

Abstract

We introduce the concept of 3-Lie-Rinehart superalgebra and systematically describe a cohomology complex by considering coefficient modules. Furthermore, we study the relationships between a Lie-Rinehart superalgebra and its induced 3-Lie-Rinehart superalgebra. The deformations of 3-Lie-Rinehart superalgebra are considered via the cohomology theory.

Split 3-Lie-Rinehart color algebras

“…Definition 2.4. [4] Let (L, [., ., . ], ǫ) be a 3−Lie color algebra and V be a G−graded vector space and ρ : L × L −→ gl(V ) be a even linear mapping.…”

“…], ǫ) be a 3−Lie color algebra and V be a G−graded vector space and ρ : L × L −→ gl(V ) be a even linear mapping. Then (V, ρ) is called a representation of L or V is an L−module if the following two conditions are satisfied: (3) if G = Z 2 and ǫ(x, y) = (−1) xy for any homogeneous elements x, y ∈ L. We then get the notion of 3−Lie-Rinehart superalgebra (see [4]).…”

“…In this paper, we continue to study the structure of a generalization of Lie-Rinehart superalgebras to the class which are called split 3−Lie-Rinehart color algebras. The definition, basic structures, actions and crossed modules of 3−Lie-Rinehart algebras and also 3−Lie-Rinehart superalgebras can be fond in [4,6]. A 3−Lie-Rinehart color algebra is a triple (L, A, ρ), where L is a 3−Lie color algebra, A is a commutative, associative color algebra, L is an A−module, (A, ρ) is a 3−Lie color algebra L−module and ρ(L, L) ⊂ Der(A).…”

Extensions and crossed modules of $n$-Lie Rinehart algebras

“…Lie Rinehart algebras have been investigated further in many papers [10,12,14,15,19,21,33,36]. Some generalizations of Lie Rinehart algebras, such as Lie Rinehart superalgebras [20] or restricted Lie Rinehart algebras [23] or 3-Lie Rinehart superalgebras [9] have been recently studied.…”