An inverse semigroup approach to self-similar k-graph $C^*$-algebras and simplicity

Abstract: We generalize the Li-Yang notion of self-similar $k$-graph $(G,\Lambda)$ and its $C<sup>*$-algebra $\mathcal{O}<em>{G,\Lambda}$ to any finitely aligned $k$-graph $\Lambda$. We then introduce an inverse semigroup model for $\mathcal{O}</em>{G,\Lambda}$ and analyze its tight groupoid and $C<sup>*$-algebra via inverse semigroup methods.