On transposed Poisson conformal superalgebras

TL;DR

This paper introduces and explores transposed Poisson conformal superalgebras, establishing their properties, relationships with other algebraic structures, and classifying structures on rank (1+1) superalgebras.

Contribution

It defines transposed Poisson conformal superalgebras, investigates their properties, and classifies all such structures on rank (1+1) superalgebras.

Findings

Tensor product of two superalgebras retains a transposed Poisson conformal structure

Established a relationship with Hom-Lie conformal superalgebras

Classified structures on rank (1+1) superalgebras

Abstract

We introduce and study transposed Poisson conformal superalgebras, the $\mathbb Z_2$-graded conformal analogues of transposed Poisson algebras, as well as their noncommutative variants. We derive a family of identities forced by the transposed conformal super-Leibniz rule and prove that the tensor product over $\mathbb C[\partial]$ of two such superalgebras again carries a natural transposed Poisson conformal superalgebra structure. Moreover, we display a close relationship between transposed Poisson conformal superalgebras and Hom-Lie conformal superalgebras, and give the compatibility conditions between a Poisson conformal superalgebra and a transposed Poisson conformal superalgebra. In addition, several constructions are obtained from modified Lie conformal brackets and from Novikov-Poisson, pre-Lie commutative, differential Novikov-Poisson, and pre-Lie Poisson conformal superalgebras. Finally, using the known classification of Lie conformal superalgebras of rank (1+1), we determine all compatible transposed Poisson conformal superalgebra structures on such superalgebras.