Operadic approach to HNN-extensions of Leibniz algebras

TL;DR

This paper develops an operadic framework for HNN-extensions of Leibniz algebras, providing presentations, bases, and embedding results, thereby advancing the algebraic theory of di-algebras.

Contribution

It introduces a novel operadic approach to HNN-extensions of Leibniz algebras, including explicit presentations and embedding theorems.

Findings

Constructed HNN-extensions of Lie di-algebras within di-algebras.

Provided a presentation for the replicated HNN-extension.

Proved Lie di-algebras embed into their HNN-extensions.

Abstract

We construct HNN-extensions of Lie di-algebras in the variety of di-algebras and provide a presentation for the replicated HNN-extension of a Lie di-algebras. Then, by applying the method of Gr\"obner-Shirshov bases for replicated algebras, we obtain a linear basis. As an application of HNN-extensions, we prove that Lie di-algebras are embedded in their HNN-extension.