Irreducible representations of Leavitt algebras

2021/03/22 by Roozbeh Hazrat, Hazrat, Roozbeh, Raimund Preusser +3
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Operator Algebras (math.OA) #Representation Theory (math.RT) #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.2103.11700

openalex publication_date 2021/03/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a weighted graph E, we construct representation graphs F, and consequently, L_K(E)-modules V_F, where L_K(E) is the Leavitt path algebra associated to E, with coefficients in a field K. We characterise representation graphs F such that V_F are simple L_K(E)-modules. We show that the category of representation graphs of E, RG(E), is a disjoint union of subcategories, each of which contains a unique universal object T which gives an indecomposable L_K(E)-module V_T and a unique irreducible representation graph S, which gives a simple L_K(E)-module V_S. Specialising to graphs with one vertex and m loops of weight n, we construct irreducible representations for the celebrated Leavitt algebras L_K(n,m). On the other hand, specialising to graphs with weight one, we recover the simple modules of Leavitt path algebras constructed by Chen via infinite paths or sinks and give a large class of non-simple indecomposable modules.

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