Quadratic Lie conformal superalgebras related to Novikov superalgebras

TL;DR

This paper explores quadratic Lie conformal superalgebras linked to Novikov superalgebras, constructing enveloping superalgebras and proving finite faithful representations for certain cases, advancing understanding of their structure and representations.

Contribution

It introduces a method to construct enveloping differential Poisson superalgebras from Novikov superalgebras and proves finite faithful representations exist for quadratic Lie conformal superalgebras based on special Gelfand--Dorfman superalgebras.

Findings

Construction of enveloping differential Poisson superalgebras from Novikov superalgebras

Proof that quadratic Lie conformal superalgebras on finite-dimensional special Gelfand--Dorfman superalgebras have finite faithful representations

Progress towards solving the open problem on finite faithful conformal representations of finite Lie conformal superalgebras

Abstract

We study quadratic Lie conformal superalgebras associated with No\-vikov 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 Gel'fand--Dorfman superalgebra has a finite faithful conformal representation. This statement is a step toward a solution of the following open problem: whether a finite Lie conformal (super)algebra has a finite faithful conformal representation.