Structure of diagonal-invariant ideals in Exel-Pardo $*$-agebras

As a unified treatment of Katsura and Nekrashevych $C^*$-algebras, Exel and Pardo introduced self-similar graph $C^*$-algebras in 2017. More recently, the algebraic version of these $C^*$-algebras (called Exel-Pardo algebras) are introduced and considered by some authors. In this note, we study the ideal structure of Exel-Pardo algebras. To do this, we first give a short proof for representing these algebras as Steinberg algebras. Then, by this result, we characterize basic, graded, and diagonal-invariant ideals of Exel-Pardo algebras by underlying graph structure. This result generalizes the graded ideal structure of Leavitt path algebras to self-similar graphs.