algebraic cone; monoid action; group action; orbit; action with unique fixed point; algebra automorphism; stable ideal; irreducible algebraic variety Cluster algebra; Kac-Moody group; Kac-Moody algebra; Semicanonical basis; Frobenius category; Preprojective algebra; Representation theory; Unipotent group; Algebraic group; Calabi-Yau category; Quasihereditary algebra 