Conformal representations of Leibniz algebras

TL;DR

This paper provides a detailed exposition of conformal representations of Leibniz algebras, showing that finite-dimensional Leibniz algebras can be faithfully represented conformally, leading to an analogue of the PBW theorem.

Contribution

It introduces explicit conformal representations for Leibniz algebras and proves their faithfulness, extending the PBW theorem to this setting.

Findings

Finite-dimensional Leibniz algebras have faithful conformal representations.

Embedded Leibniz algebras into conformal algebras of the same variety.

Established an analogue of the PBW theorem for Leibniz algebras.

Abstract

In this note we present a more detailed and explicit exposition of the definition of a conformal representation of a Leibniz algebra. Recall (arXiv:math/0611501v3) that Leibniz algebras are exactly Lie dialgebras. The idea is based on the general fact that every dialgebra that belongs to a variety $\Var $ can be embedded into a conformal algebra of the same variety. In particular, we prove that an arbitrary (finite dimensional) Leibniz algebra has a (finite) faithful conformal representation. As a corollary, we deduce the analogue of the PBW-theorem for Leibniz algebras.