On the structure and representations of the insertion-elimination Lie algebra

TL;DR

This paper investigates the structure of the insertion-elimination Lie algebra on rooted trees, establishing its simplicity, lack of finite-dimensional representations, and classifying its irreducible lowest-weight modules.

Contribution

It proves the algebra's simplicity, shows it has no finite-dimensional representations, and classifies its irreducible lowest-weight modules.

Findings

The algebra is simple.

It has no finite-dimensional representations.

Irreducible modules are classified by lowest weights.

Abstract

We examine the structure of the insertion-elimination Lie algebra on rooted trees introduced in \cite{CK}. It possesses a triangular structure $\g = \n_+ \oplus \mathbb{C}.d \oplus \n_-$, like the Heisenberg, Virasoro, and affine algebras. We show in particular that it is simple, which in turn implies that it has no finite-dimensional representations. We consider a category of lowest-weight representations, and show that irreducible representations are uniquely determined by a "lowest weight" $\lambda \in \mathbb{C}$. We show that each irreducible representation is a quotient of a Verma-type object, which is generically irreducible.