Work

The graded classification conjectures hold for various finite representations of Leavitt path algebras

Year: 2025

Type: article

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 K 0 gr ( L k ( E ) ) ≅ K 0 gr ( L k ( F ) ) as order-preserving 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 no sinks and sources, an order-preserving Z [ x , x − 1 ] -module isomorphism K 0 gr ( L k ( E ) ) ≅ K 0 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., g r Z − L k ( E ) ≈ gr g r Z − L k ( F ) . We further obtain that the category of finite dimensional (graded) modules is equivalent, i.e., mod- L k ( E ) ≈ mod- L k ( F ) and gr- L k ( E ) ≈ gr gr- L k ( F ) .

Source: Journal of Algebra

DOI: 10.1016/j.jalgebra.2025.02.035

Authors: Wolfgang Bock, Roozbeh Hazrat, Alfilgen Sebandal

Institutions: Linnaeus University, Western Sydney University, Central Visayan Institute Foundation

Language: English

Topic: Rings, Modules, and Algebras

Subfield: Algebra and Number Theory

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,310

Funder: Australian Research Council