[Paper Review] Leavitt path algebras with at most countably many irreducible representatios

Pere Ara, Kulumani M. Rangaswamy|arXiv (Cornell University)|Sep 30, 2013

Advanced Operator Algebra Research13 references3 citations

TL;DR

This paper establishes necessary and sufficient conditions for a Leavitt path algebra $L_K(E)$ to have at most countably many isomorphism classes of simple left or right modules—designated as CIRT (countable irreducible representation type). It proves that $L_K(E)$ is CIRT if and only if it is the union of a smooth ascending chain of graded ideals of countable length, where each successive quotient is a direct sum of at most countably many matrix rings over $K$ or $K[x,x^{-1}]$, and shows that the field $K$ must be countable if $E$ contains cycles.

ABSTRACT

Let E be an arbitrary directed graph with no restrictions on the number of vertices and edges and let K be any field. We give necessary and sufficient conditions for the Leavitt path algebra L_K(E) to be of countable irreducible representation type, that is, we determine when L_K(E)has at most countably many distinct isomorphism classes of simple left L_K(E-modules. It is also shown that L_K(E) has dinitely many isomorphism classes of simple left modules if and only if L_K(E) is a semi-artinian von Neumann regular ring with at most finitely many ideals. Equivalent conditions on the graph E are also given. Examples are constructed showing that for each (finite or infinite) cardinal m there exists a Leavitt path algebra L having exactly m distinct isomorphism classes of simple left modules.

Motivation & Objective

• To determine when a Leavitt path algebra $L_K(E)$ has at most countably many isomorphism classes of simple modules.
• To clarify the structural conditions on the graph $E$ and field $K$ that ensure countable irreducible representation type (CIRT).
• To establish a characterization of finite irreducible representation type via semi-artinian von Neumann regular rings with finitely many ideals.
• To construct examples of Leavitt path algebras with exactly $\kappa$ distinct isomorphism classes of simple right modules for any cardinal $\kappa$.

Proposed method

• Use of a smooth ascending chain of graded ideals $0 = I_0 < I_1 < \cdots < I_\alpha < \cdots$ with $\alpha < \tau$ for a countable ordinal $\tau$, to decompose $L_K(E)$.
• Analysis of successive quotients $I_{\alpha+1}/I_\alpha$ as direct sums of matrix rings over $K$ and $K[x,x^{-1}]$, which classify simple modules.
• Application of the theory of algebraic branching systems and Chen’s representations via sinks and infinite paths to analyze simple modules.
• Use of transfinite induction to construct the 'Pyramid' graph $P_\lambda$ for each ordinal $\lambda$, yielding $L_K(P_\lambda)$ with exactly $|\lambda|$ isomorphism classes of simple modules.
• Leveraging the structure of line points and hereditary saturated sets in $E$ to control the module category.
• Establishing that $L_K(E)$ is semi-artinian von Neumann regular with finitely many ideals if and only if it has finitely many simple modules.

Experimental results

Research questions

• RQ1 What conditions on the graph $E$ and field $K$ ensure that $L_K(E)$ has at most countably many isomorphism classes of simple left modules?
• RQ2 When does $L_K(E)$ have only finitely many isomorphism classes of simple modules, and how is this related to ring-theoretic properties like semi-artinianism and von Neumann regularity?
• RQ3 Can Leavitt path algebras realize any given cardinality $\kappa$ as the number of isomorphism classes of simple right modules, and how is this achieved?
• RQ4 How does the presence of cycles in $E$ constrain the field $K$ in the context of CIRT algebras?
• RQ5 What role do graded ideals and their successive quotients play in classifying the simple module categories of $L_K(E)$?

Key findings

• A Leavitt path algebra $L_K(E)$ is of countable irreducible representation type (CIRT) if and only if it is the union of a smooth ascending chain of graded ideals of countable length, where each quotient $I_{\alpha+1}/I_\alpha$ is a direct sum of at most countably many matrix rings over $K$ or $K[x,x^{-1}]$.
• If the graph $E$ contains cycles, then the field $K$ must be countable for $L_K(E)$ to be CIRT.
• The algebra $L_K(E)$ has finitely many isomorphism classes of simple left/right modules if and only if it is a semi-artinian von Neumann regular ring with finitely many ideals.
• For any cardinal $\kappa$, there exists a Leavitt path algebra $L_K(E)$ with exactly $\kappa$ distinct isomorphism classes of simple right $L_K(E)$-modules.
• The construction of the 'Pyramid' graph $P_\lambda$ via transfinite induction yields $L_K(P_\lambda)$ with exactly $|\lambda|$ isomorphism classes of simple modules.
• When $E$ is finite, $L_K(E)$ has finitely many simple modules if and only if $L_K(E)$ is artinian semisimple, which occurs precisely when $E$ is acyclic.