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

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.

References in corpus (2)

• Insertion and Elimination Lie Algebra: the Ladder case
• The Structure of the Ladder Insertion-Elimination Lie algebra

Cited by in corpus (1)

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