3-Lie-Rinehart Algebras

Abstract

In this paper, we define a class of 3-algebras which are called 3-Lie-Rinehart algebras. A 3-Lie-Rinehart algebra is a triple $(L,A,\rho )$, where $A$ is a commutative associative algebra, $L$ is an $A$-module, $(A,\rho )$ is a 3-Lie algebra $L$-module and $\rho (L,L)\subseteq Der(A)$. We discuss the basic structures, actions and crossed modules of 3-Lie-Rinehart algebras and construct 3-Lie-Rinehart algebras from given algebras, we also study the derivations from 3-Lie-Rinehart algebras to 3-Lie $A$-algebras. From the study, we see that there is much difference between 3-Lie algebras and 3-Lie-Rinehart algebras.

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Cited by in corpus (12)

• Cohomologies and crossed modules for pre-Lie Rinehart algebras
• Universal enveloping algebras of Lie-Rinehart algebras as a left adjoint functor
• Extensions and crossed modules of $n$-Lie Rinehart algebras
• On split regular Hom-Leibniz-Rinehart algebras
• Hom 3-Lie-Rinehart Algebras
• Convolution bialgebra of a Lie groupoid and transversal distributions
• Fiber-Wise Linear Differential Operators
• Extension, deformation and categorification of $\text{AssDer}$ pairs
• Split 3-Lie-Rinehart color algebras
• On 3-Hom-Lie-Rinehart algebras
• Generalized Lie Algebroids - Examples by Distinguished Lie Algebroids with Applications to Optimal Control
• Batalin--Vilkovisky algebra structures on the Hochschild cohomology of generalized Weyl algebras