Odd and even derivations, transposed Poisson superalgebra and 3-Lie superalgebra
Viktor Abramov, Nikolai Sovetnikov
TL;DR
The paper addresses extending transposed Poisson algebra concepts to odd derivations in superalgebras. It introduces a TP-compatible framework on a left $A$-supermodule with a Jordan structure to accommodate odd derivations, and provides an explicit construction ${\mathfrak D}^{\delta}$ that realizes this structure; it also shows that, for even derivations, a transposed Poisson superalgebra grants a graded ternary bracket that yields a 3-Lie superalgebra. The main contributions are the formalization of TP-compatible left $A$-supermodules, the odd-derivation example, and the graded generalization of the Bai–Guo–Wu 3-Lie construction to the super setting. This work broadens the algebraic toolbox for transposed Poisson structures in supergeometry and connects to higher-arity Lie-type algebras with potential applications in graded Hamiltonian frameworks.
Abstract
One important example of a transposed Poisson algebra can be constructed by means of a commutative algebra and its derivation. This approach can be extended to superalgebras, that is, one can construct a transposed Poisson superalgebra given a commutative superalgebra and its even derivation. In this paper we show that including odd derivations in the framework of this approach requires introducing a new notion. It is a super vector space with two operations that satisfy the compatibility condition of transposed Poisson superalgebra. The first operation is determined by a left supermodule over commutative superalgebra and the second is a Jordan bracket. Then it is proved that the super vector space generated by an odd derivation of a commutative superalgebra satisfies all the requirements of introduced notion. We also show how to construct a 3-Lie superalgebra if we are given a transposed Poisson superalgebra and its even derivation.