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A note on Lie and Jordan structures of Leavitt path algebras

Huynh Viet Khanh, Le Qui Danh

TL;DR

The paper determines exactly when the Lie algebra $\mathbf{K}_{L_K(E)}$ and the Jordan algebra $\mathbf{S}_{L_K(E)}$ (arising from the Leavitt path algebra $L_K(E)$ with the standard involution) are solvable, building on Herstein’s foundational work and recent classifications. It develops a matrix-algebra with involution framework to derive precise criteria based on the matrix size $n$, the field characteristic, and whether the involution is trivial, then translates these into graph-theoretic conditions for Leavitt path algebras by decomposing $L_K(E)$ into matrix-like ideals. The main contributions are exact solvability and nilpotence indices for various cases: for $\mathbf{K}_{\mathbb{M}_n(\mathcal{A})}$, solvability hinges on $n$ and $(\mathrm{char}(K),\text{involution})$; for $L_K(E)$, solvability of $\mathbf{K}_{L_K(E)}$ (char $2$) occurs precisely when $E$ is a disjoint union of the graphs $E_1$–$E_6$, with index bounds, and (char $\neq2$) when $E$ is a disjoint union of $E_1,E_2,E_4$; analogous criteria are given for the Jordan solvable $\mathbf{S}_{L_K(E)}$ and related structures. These results provide sharp, graph-theoretic criteria for Lie and Jordan solvability in Leavitt path algebras with involution.

Abstract

Let $L_K(E)$ be the Leavitt path algebra of a directed graph $E$ over a field $K$. In this paper, we determine $E$ and $K$ for the Lie algebra $\mathbf{K}_{L_K(E)}$ and the Jordan algebra $\mathbf{S}_{L_K(E)}$ arising from $L_K(E)$ with respect to the standard involution to be solvable.

A note on Lie and Jordan structures of Leavitt path algebras

TL;DR

The paper determines exactly when the Lie algebra and the Jordan algebra (arising from the Leavitt path algebra with the standard involution) are solvable, building on Herstein’s foundational work and recent classifications. It develops a matrix-algebra with involution framework to derive precise criteria based on the matrix size , the field characteristic, and whether the involution is trivial, then translates these into graph-theoretic conditions for Leavitt path algebras by decomposing into matrix-like ideals. The main contributions are exact solvability and nilpotence indices for various cases: for , solvability hinges on and ; for , solvability of (char ) occurs precisely when is a disjoint union of the graphs –, with index bounds, and (char ) when is a disjoint union of ; analogous criteria are given for the Jordan solvable and related structures. These results provide sharp, graph-theoretic criteria for Lie and Jordan solvability in Leavitt path algebras with involution.

Abstract

Let be the Leavitt path algebra of a directed graph over a field . In this paper, we determine and for the Lie algebra and the Jordan algebra arising from with respect to the standard involution to be solvable.

Paper Structure

This paper contains 7 sections, 14 theorems, 40 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Directed graphs and Leavitt path algebras.
  4. Lie and Jordan structures of associative algebras
  5. Homomorphisms of algebras with involutions
  6. Lie Solvability and Jordan Solvability of $\mathbb{M}_n(\mathcal{A})$
  7. Lie Solvability and Jordan Solvability of $L_K(E)$

Key Result

Proposition 3.1

Let $K$ be a field, and $\mathcal{A}$ be a unital $K$-algebra with an involution $^\natural$. Assume further that $\mathcal{A}$ is an integral domain. Let $\mathbf{K}_{\mathbb{M}_n(\mathcal{A})}$ be as defined above. Then, the following assertions holds:

Figures (1)

  • Figure :

Theorems & Definitions (26)

  • Definition 2.1: Leavitt path algebra
  • Proposition 3.1
  • proof
  • Corollary 3.2
  • Corollary 3.3
  • Lemma 4.1: Bo_abrams_2017
  • Lemma 4.2
  • proof
  • Remark 1
  • Remark 2
  • ...and 16 more