[Paper Review] Morita Equivalence of Graph and Ultragraph Leavitt Path Algebras

Michael Mekonen Firrisa|arXiv (Cornell University)|Jun 11, 2020

Advanced Operator Algebra Research22 references4 citations

TL;DR

This paper establishes that every ultragraph Leavitt path algebra over a field is Morita equivalent to a graph Leavitt path algebra, providing an algebraic analog to a known result in C*-algebras. The key contribution is a Morita equivalence that enables an alternative proof of the simplicity criterion for ultragraph Leavitt path algebras using desingularization of ultragraphs.

ABSTRACT

The primary purpose of this thesis is to show every ultragraph Leavitt path algebra is Morita equivalent, as a ring, to a graph Leavitt path algebra. Takeshi Katsura, Paul Muhly, Aidan Sims, and Mark Tomforde showed every ultragraph $C^{*}$-algebra is Morita equivalent, in the $C^{*}$-sense, to a graph $C^{*}$-algebra; our result is an algebraic analog of this fact. Further, we will use our result to give an alternate proof for established conditions which guarantee the simplicity of an ultragraph Leavitt path algebra over a field.

Motivation & Objective

• To establish Morita equivalence between ultragraph Leavitt path algebras and graph Leavitt path algebras over a field.
• To provide an algebraic analog of the C*-algebraic result that every ultragraph C*-algebra is Morita equivalent to a graph C*-algebra.
• To use this equivalence to give an alternative proof of the simplicity criterion for ultragraph Leavitt path algebras.
• To extend the understanding of ideal structures in ultragraph Leavitt path algebras via desingularization techniques.

Proposed method

• Constructing a desingularization of an ultragraph to produce a graph whose Leavitt path algebra is Morita equivalent to the original ultragraph algebra.
• Proving that the desingularized graph satisfies Condition (L) if and only if the original ultragraph does.
• Using the fact that Morita equivalence preserves simplicity to transfer the simplicity criterion from graph to ultragraph Leavitt path algebras.
• Leveraging the known simplicity theorem for graph Leavitt path algebras to derive the corresponding result for ultragraphs.
• Establishing a correspondence between saturated hereditary subsets in the ultragraph and those in its desingularized graph.
• Utilizing the lattice isomorphism of ideals between the original and desingularized algebras to preserve structural properties.

Experimental results

Research questions

• RQ1 Is every ultragraph Leavitt path algebra Morita equivalent to a graph Leavitt path algebra?
• RQ2 Can the Morita equivalence between ultragraph and graph C*-algebras be lifted to the algebraic setting of Leavitt path algebras?
• RQ3 Does the simplicity of an ultragraph Leavitt path algebra correspond to Condition (L) and maximality of saturated hereditary subsets in the ultragraph?
• RQ4 Can the simplicity criterion for ultragraph Leavitt path algebras be re-derived using Morita equivalence and desingularization?

Key findings

• Every ultragraph Leavitt path algebra over a field is Morita equivalent to a graph Leavitt path algebra.
• The desingularization process preserves the essential graph-theoretic properties needed for Morita equivalence, including Condition (L).
• The simplicity of an ultragraph Leavitt path algebra is equivalent to the ultragraph satisfying Condition (L) and having only trivial saturated hereditary subsets.
• The Morita equivalence allows an alternative proof of the simplicity criterion for ultragraph Leavitt path algebras, relying on the known result for graph algebras.
• The lattice of ideals in the ultragraph algebra corresponds to that in the desingularized graph algebra, preserving the structure of basic ideals.