Automorphisms and derivations of the insertion-elimination algebra and related graded Lie algebras

TL;DR

This paper investigates the structural properties of the insertion-elimination algebra, including automorphisms, derivations, and subalgebras, and extends some results to related graded Lie algebras and generalized Virasoro algebras.

Contribution

It provides a detailed analysis of the automorphism and derivation groups of the insertion-elimination algebra and generalizes some results to broader classes of Lie algebras.

Findings

Finite-dimensional subalgebras characterized

Automorphism and derivation groups determined

Generating sets for the algebra identified

Abstract

This paper addresses several structural aspects of the insertion-elimination algebra, a Lie algebra that can be realized in terms of tree-inserting and tree-eliminating operations on the set of rooted trees. In particular, we determine the finite-dimensional subalgebras, the automorphism group, the derivation group, and a generating set for the insertion-elimination Lie algebra. Many parts of the results are stated for a more general class of Lie algebras and reproduce results for the generalized Virasoro algebras.