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Tanaka rigidity of graph Lie algebras

Mauricio Godoy Molina

Abstract

We give sufficient conditions on a labeled direct graph to determine whether the Tanaka prolongation of its associated Lie algebra is infinite-dimensional. In the case that all directed edges are labeled differently, the corresponding graph Lie algebra is of infinite type if and only if the graph has a vertex of degree one.

Tanaka rigidity of graph Lie algebras

Abstract

We give sufficient conditions on a labeled direct graph to determine whether the Tanaka prolongation of its associated Lie algebra is infinite-dimensional. In the case that all directed edges are labeled differently, the corresponding graph Lie algebra is of infinite type if and only if the graph has a vertex of degree one.
Paper Structure (2 sections, 4 theorems, 23 equations)

This paper contains 2 sections, 4 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Tanaka rigidity for graph Lie algebras

Key Result

Proposition 2.1

Assume $G$ is a connected graph satisfying hyp:H. Then where ${\mathcal{Z}}({\rm Lie}(G))$ denotes the center of the graph Lie algebra ${\rm Lie}(G)$.

Theorems & Definitions (9)

  • Proposition 2.1
  • proof
  • Theorem 2.2
  • proof
  • Proposition 2.3
  • proof
  • Theorem 2.4
  • proof
  • Remark 2.5