Leavitt Path Algebras of Quantum Quivers

TL;DR

This paper introduces Quantum Quivers, a novel algebraic structure replacing graph vertices and edges with $C^*$-algebras and $*$-homomorphisms, and develops their associated Leavitt path algebras.

Contribution

It extends the concept of Leavitt path algebras to Quantum Quivers, providing new theoretical foundations and module classification results.

Findings

Defined Quantum Quivers as $C^*$-algebraic analogues of quivers

Constructed Leavitt path algebras over Quantum Quivers

Computed the monoid of finitely generated projective modules

Abstract

Adapting a recent work of Brannan et al., on extending graph $C^*$-algebras to Quantum graphs, we introduce "Quantum Quivers" as an analogue of quivers where the edge and vertex set has been replaced by a $C^*$-algebra and the maps between the sets by $*$-homomorphisms. Additionally, we develop the theory around these structures and construct a notion of Leavitt path algebra over them and also compute the monoid of finitely generated projective modules over this class of algebras.