Whittaker modules for the insertion-elimination Lie algebra

TL;DR

This paper studies the representation theory of the insertion-elimination Lie algebra, focusing on Whittaker modules, their construction, and conditions for simplicity, within the context of rooted tree operations.

Contribution

It introduces the concept of Whittaker modules for the insertion-elimination Lie algebra and establishes conditions under which these modules are simple.

Findings

Standard Whittaker modules are simple under certain algebra homomorphism constraints.

The insertion-elimination algebra admits a triangular decomposition.

Representation theory insights for the insertion-elimination Lie algebra.

Abstract

This paper addresses the representation theory of the insertion-elimination Lie algebra, a Lie algebra that can be naturally realized in terms of tree-inserting and tree-eliminating operations on rooted trees. The insertion-elimination algebra admits a triangular decomposition in the sense of Moody and Pianzola, and thus it is natural to define a Whittaker module corresponding to a given algebra homomorphism. Among other results, we show that the standard Whittaker module is simple given certain constraints on the corresponding algebra homomorphism.