Diagrams of representations

TL;DR

This paper introduces a diagrammatic approach to analyze Lie algebra representations, focusing on normal forms, orbits, and invariants, particularly for nilpotent Lie algebras, by translating algebraic properties into directed graph structures.

Contribution

It presents a novel method of using diagrams to study representations of Lie algebras, facilitating the understanding of their structure and invariants.

Findings

Diagrams effectively describe normal forms of representations.

The approach simplifies the analysis of orbits in Lie algebra representations.

Invariants can be characterized using the diagrammatic framework.

Abstract

For a representation of a Lie algebra, one can construct a diagram of the representation, i. e. a directed graph with edges labeled by matrix elements of the representation. This article explains how to use these diagrams to describe normal forms, orbits and invariants of the representation, especially for the case of nilpotent Lie algebras.