On transposed Poisson conformal 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.

23 pages, no figures