Algebraic groups and their tangent spaces

Abstract

This document is the culmination of an undergraduate final-year project, undertaken in the academic year of 2021-2022. The goal of this project was to examine how Lie groups and their corresponding Lie algebras can be realised as algebro-geometric objects. Additionally, my supervisor's aim was also for me to gain some experience and intuition with classical algebraic geometry. We used `An Introduction to Algebraic Geometry and Algebraic Groups' by Geck as the primary reference. This report develops the theory as explicitly as possible to introduce the reader to ideas in classical algebraic geometry, and how they can be used to construct Lie groups and Lie algebras.