algebraic cone; monoid action; group action; orbit; action with unique fixed point; algebra automorphism; stable ideal; irreducible algebraic variety algebraic groups; irreducible representations; Hopf algebras; Lie algebras; unipotent algebraic groups; tensor product; semisimple group representations; varieties; dimension theory of local rings; tangent spaces; Borel subgroups; Galois cohomology; automorphism groups; weights; universal enveloping algebra 