algebraic cone; monoid action; group action; orbit; action with unique fixed point; algebra automorphism; stable ideal; irreducible algebraic variety derived category; Mukai flop; non-commutative crepant resolution; (NCCR); quiver representation; Gorenstein algebra; tilting object; tilting bundle; total spaces; double Beilinson quiver; path algebra; McKay correspondence; group representaitons; geometric resolution; algebraic resolution; multi-mutation functors 