An inverse semigroup approach to self-similar k-graph $C^{\ast }$-algebras and simplicity

We generalize the Li-Yang notion of self-similar $k$-graph $(G,\mathrm{\Lambda })$ and its $C^{\ast }$-algebra ${\mathcal{O}}_{G,\mathrm{\Lambda }}$ to any finitely aligned $k$-graph $\mathrm{\Lambda }$. We then introduce an inverse semigroup model for ${\mathcal{O}}_{G,\mathrm{\Lambda }}$ and analyze its tight groupoid and $C^{\ast }$-algebra via inverse semigroup methods.