Unipotent Algebraic and Geometric Groups: ARefined Exposition with Detailed Proofs and Original Results

Abstract

We present a refined exposition of unipotent algebraic and geometric groups by streamlining classical sections and providing complete, detailed proofs of the fundamental theorems. In addition to covering definitions, properties, classification via Lie algebras, and structural aspects (including filtrations and central series), we contribute an original result on the geometric structure of orbits under unipotent group actions. In particular, we prove that the orbits of a connected unipotent group acting on an irreducible affine variety are isomorphic to affine spaces. This work is intended for researchers and advanced graduate students in algebraic groups and algebraic geometry