Quantum Cuntz—Krieger algebras

Given a classical simple graph $G$, there is a $\{0,1\}$-matrix $A_G$ which represents which vertices are connected via a directed edge. This matrix gives rise to the Cuntz—Krieger algebra $\mathcal{O}_{A_G}$. Quantum graphs have different (though equivalent) definitions in recent literature; our presentation of them in this talk is as a {\em quantum} adjacency matrix on a {\em quantum} set, which generalizes the classical $\{0,1\}$-adjacency matrix acting as a linear operator on a complex vector space whose dimension is the size of the underlying graph's vertex set. Analogous to the classical setting, we can associate a quantum Cuntz—Krieger (QCK) algebra. There is an additional structure called a local quantum Cuntz—Krieger algebra, which is a guaranteed quotient of the QCK algebra. In this talk, we will discuss the challenges of studying the (full) quantum Cuntz--Krieger algebra, and pose the question of whether or not there is quantum graph for which the local and full QCK algebras are distinct. Finally, we present a method for generating examples of quantum graphs whose full and local QCK algebras are distinct.