algebraic cone; monoid action; group action; orbit; action with unique fixed point; algebra automorphism; stable ideal; irreducible algebraic variety Hopf algebras; triangular \(n \times n\) matrices; group valued functor; comultiplication; bigebra; antipode; anti-homomorphism; generators; Hopf ideals; deformations; unipotent algebraic group schemes; quantum universal enveloping algebra 