Universal enveloping algebra

A universal enveloping algebra is a mathematical construction that translates the structure of a Lie algebra—an algebraic system capturing the notion of symmetry and infinitesimal transformations—into an associative algebra, where elements can be multiplied without restrictions. It serves as a bridge allowing us to work within an environment where tools for associative algebras apply, facilitating the study of Lie algebras through representations and algebraic techniques. Essentially, it 'envelops' the Lie algebra in a larger, more flexible structure that preserves its properties while enabling broader algebraic operations.