Structure and cohomology of 3-Lie-Rinehart superalgebras

Abstract

We introduce a 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 a cohomology theory.

Graded Lie-Rinehart Algebras

“…Let consider now the fourth summand in (7), the bilinearity of the product and Jacobi identity lead to…”

“…Therefore all summands in (7) are contained in L [g],1 , and therefore all summands in (6) are included in…”

“…have studied thoroughly Lie-Rinehart structures in connection with symplectic geometry, Poisson structures, Lie groupoids and algebroids and other types of quantizations [17,[25][26][27]32,33]. Throughout the last years have been studied also some generalizations of these algebras such as restricted Lie-Rinehart algebras [14] or 3-Lie-Rinehart superalgebras [7]. Consequently, one can assert that Lie-Rinehart algebras constitutes a fundamental tool in many topics of research in mathematics.…”