semi-simple Lie algebra

A semi-simple Lie algebra is a mathematical structure used to study continuous symmetries, such as rotations or transformations in space. It can be thought of as a system composed of interconnected, non-abelian building blocks called simple Lie algebras, combined without any "degenerate" or abelian parts (which commute). Semi-simple Lie algebras are important because they have well-understood classification, allowing mathematicians and physicists to analyze complex symmetry behaviors systematically. They underpin many areas, including particle physics, geometry, and the theory of differential equations, helping us understand the fundamental symmetry properties of various systems.