Jornal de Teoria Generalizada de Mentiras e Aplicações

Exploring Lie Theory: Symmetries in Mathematics and Physics

Abstract

Timothy Levine

Lie theory, also known as the theory of Lie groups, is a branch of mathematics that studies continuous symmetry, especially in the context of smooth manifolds. Lie theory provides a mathematical framework to describe the symmetries of physical systems, from the motion of planets to the behavior of subatomic particles. The theory also plays a crucial role in many areas of modern mathematics, including differential geometry, representation theory, algebraic geometry and mathematical physics. This article provides an introduction to Lie theory, starting with the basics of Lie groups and Lie algebras and progressing to more advanced topics such as Lie's third theorem and the classification of semisimple Lie algebras.