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October 9, 2025Open Access

Odd and even derivations, transposed Poisson superalgebra and 3-Lie superalgebra

Key Points

Introducing odd derivations necessitates a new super vector space concept for transposed Poisson superalgebras, expanding existing frameworks.
The study provides proof that super vector spaces generated by odd derivations comply with the compatibility requirements outlined in the new definition.
It shows that given a transposed Poisson superalgebra, one can construct a 3-Lie superalgebra through its even derivation, enriching algebraic structures.
Key operations involve a left supermodule and a Jordan bracket, shedding light on the relationships between these algebraic entities.

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 a transposed Poisson superalgebra. The first operation is determined by a left supermodule over a 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 the introduced notion. We also show how to construct a 3-Lie superalgebra if we are given a transposed Poisson superalgebra and its even derivation.

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