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On the homogeneous zero components of Leavitt algebras

Raimund Preusser

TL;DR

This work describes the zero component $L(m,n)_0$ of the Leavitt algebra $L(m,n)$ under the canonical $\mathbb{Z}$-grading, proving $L(m,n)_0 \cong A(m,n) * A(n,m)$ where $A(m,n)$ is a direct limit of Bergman algebras. It establishes $A(m,n) \cong L(m,n)_0^{xy}$ and $A(n,m) \cong L(m,n)_0^{yx}$, with $L(m,n)_0^{xy}$ realized via a detailed basis and a presentation $A(m,n)$. For $m=1$, this recovers the classical direct limit of matrix algebras, i.e., $L(1,n)_0$ as a direct limit of matrix algebras. The paper also analyzes the graded $\mathcal{V}$-monoid, shows $\mathcal{V}(L(m,n)_0) \cong \mathcal{V}^{\mathrm{gr}}(L(m,n))$, and proves that $L(m,n)_0$ has the Invariant Basis Number property, highlighting a structural contrast with the full algebra $L(m,n)$ and broadening connections to Bergman algebras and Leavitt path algebras.

Abstract

We prove that the zero component $L(m,n)_0$ of a Leavitt algebra $L(m,n)$ with respect to the canonical grading is a direct limit $\varinjlim_{z}L(m,n)_{0,z}$, where each algebra $L(m,n)_{0,z}$ is a free product of two Bergman algebras. For the special case $m=1,n>1$, one recovers the known result that the zero component $L(1,n)_0$ is a direct limit of matrix algebras. Moreover, we show that $L(m,n)_0$ has the IBN property.

On the homogeneous zero components of Leavitt algebras

TL;DR

This work describes the zero component of the Leavitt algebra under the canonical -grading, proving where is a direct limit of Bergman algebras. It establishes and , with realized via a detailed basis and a presentation . For , this recovers the classical direct limit of matrix algebras, i.e., as a direct limit of matrix algebras. The paper also analyzes the graded -monoid, shows , and proves that has the Invariant Basis Number property, highlighting a structural contrast with the full algebra and broadening connections to Bergman algebras and Leavitt path algebras.

Abstract

We prove that the zero component of a Leavitt algebra with respect to the canonical grading is a direct limit , where each algebra is a free product of two Bergman algebras. For the special case , one recovers the known result that the zero component is a direct limit of matrix algebras. Moreover, we show that has the IBN property.
Paper Structure (19 sections, 28 theorems, 96 equations)

This paper contains 19 sections, 28 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graded algebras
  4. Words
  5. Linear transformations between countably infinite-dimensional vector spaces
  6. The monoid $\mathfrak{M}_n(R)$
  7. Leavitt algebras
  8. The algebra $L(m,n)$
  9. The canonical grading of $L(m,n)$
  10. The basis $\mathcal{B}$ for $L(m,n)_0$
  11. The free factors $L(m,n)_0^{xy}$ and $L(m,n)_0^{yx}$
  12. The basis $\mathcal{C}^{xy}$
  13. The basis $\mathcal{D}^{xy}$
  14. The main results
  15. The algebras $A(m,n)$
  16. ...and 4 more sections

Key Result

Lemma 2.1

If $\phi \colon \bigoplus_{\mathbb{N}} K \to \bigoplus_{\mathbb{N}} K$ is a linear transformation, then there is a unique matrix $A \in \operatorname{Mat}_{\mathbb{N}}(K)$ such that $\phi = \phi_A$, namely the matrix $A$ whose $j$-th column equals $\phi (e_j)$.

Theorems & Definitions (67)

  • Lemma 2.1
  • Lemma 2.2
  • Definition 2.3
  • Lemma 2.4
  • Remark 3.1
  • Proposition 3.2
  • proof
  • Theorem 3.3
  • proof
  • Lemma 3.4
  • ...and 57 more