Quadratic Lie conformal superalgebras related to Novikov superalgebras

Abstract

We study quadratic Lie conformal superalgebras associated with Novikov superalgebras. For every Novikov superalgebra $(V,\circ )$, we construct an enveloping differential Poisson superalgebra $U(V)$ with a derivation $d$ such that $u\circ v=ud(v)$ and $\{u,v\}=u\circ v-(-1{)}^{|u||v|}v\circ u$ for $u,v\in V$. The latter means that the commutator Gelfand–Dorfman superalgebra of $V$ is special. Next, we prove that every quadratic Lie conformal superalgebra constructed on a finite-dimensional special Gelfand–Dorfman superalgebra has a finite faithful conformal representation. This statement is a step towards a solution of the following open problem: whether a finite Lie conformal (super)algebra has a finite faithful conformal representation.