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On free subgroups in Leavitt path algebras

Bui Xuan Hai, Huynh Viet Khanh

Abstract

Let $E$ be a graph and $K$ a field. In this paper we prove that the multiplicative group of a unital noncommutative Leavitt path algebra $L_K(E)$ contains non-cyclic free subgroups provided $K$ is of characteristic $0$. Further, we provide a description of the generators of such free subgroups in term of the graph $E$.

On free subgroups in Leavitt path algebras

Abstract

Let be a graph and a field. In this paper we prove that the multiplicative group of a unital noncommutative Leavitt path algebra contains non-cyclic free subgroups provided is of characteristic . Further, we provide a description of the generators of such free subgroups in term of the graph .

Paper Structure

This paper contains 7 sections, 18 theorems, 60 equations, 1 figure.

Table of Contents

  1. Introduction and preliminaries
  2. Simple modules over Leavitt path algebras
  3. Non-cyclic free subgroups in Leavitt path algebras
  4. The generators of non-cyclic free subgroups
  5. Free subgroups determined by the primitive ideals of type I
  6. Free subgroups determined by the primitive ideals of type II
  7. Free subgroups determined by the primitive ideals of type III

Key Result

Lemma 2.1

Let $\mu$ be an infinite path, and $w$ a sink in $E$. Let $V_{[\mu]}$ and $\mathbf{N}_w$ be the $L_K(E)$-modules defined above. Then,

Figures (1)

  • Figure :

Theorems & Definitions (42)

  • Definition 1.1: Leavitt path algebra
  • Definition 1.2: Quotient graph
  • Remark 1.3
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • ...and 32 more