Morita theory of finite representations of Leavitt path algebras

2024/09/29 by Wolfgang Böck, Bock, Wolfgang, Roozbeh Hazrat +3 · 1 citation
Computer Science · Mathematics · #Advanced Algebra and Logic #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #K-Theory and Homology (math.KT) #Representation Theory (math.RT) #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.2409.19677

openalex publication_date 2024/09/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Graded Classification Conjecture states that for finite directed graphs E and F, the associated Leavitt path algebras L_\K(E) and L_\K(F) are graded Morita equivalent, i.e., \Gr L_\K(E) ≈_\gr \Gr L_\K(F), if and only if, their graded Grothendieck groups are isomorphic K₀^\gr(L_\K(E)) ≅ K₀^\gr(L_\K(F)) as order-preserving \mathbb Z[x,x^-1]-modules. Furthermore, if under this isomorphism, the class [L_\K(E)] is sent to [L_\K(F)] then the algebras are graded isomorphic, i.e., L_\K(E) ≅ _\gr L_\K(F). In this note we show that, for finite graphs E and F with so sinks and sources, an order-preserving \mathbb Z[x,x^-1]-module isomorphism K₀^\gr(L_\K(E)) ≅ K₀^\gr(L_\K(F)) gives that the categories of locally finite dimensional graded modules of L_\K(E) and L_\K(F) are equivalent, i.e., \fGr[ℤ] L_\K(E)≈_\gr \fGr[ℤ]L_\K(F). We further obtain that the category of finite dimensional (graded) modules are equivalent, i.e., \fModd L_\K(E) ≈ \fModd L_\K(F) and \fGr L_\K(E) ≈_\gr \fGr L_\K(F).

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• Classification and Ideal Lattices of Leavitt Path Algebras

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