PoincaréBirkhoffWitt

The Poincaré-Birkhoff-Witt theorem is a fundamental result in the study of Lie algebras. It establishes an explicit algebraic structure for the universal enveloping algebra of a Lie algebra. Specifically, it states that if g is a finite-dimensional Lie algebra over a field k, and {x1, x2, ..., xn} is a basis for g, then the universal enveloping algebra U(g) has a basis consisting of all monomials of the form x1^i1 x2^i2 ... xn^in where ij are non-negative integers. This basis provides a concrete way to understand the structure of U(g) and its representations. The theorem is named after Henri Poincaré, George Birkhoff, and Ernst Witt. Poincaré's work on differential equations and Birkhoff's on algebraic methods laid some groundwork, but it was Witt who first proved the theorem in 1937. The theorem has significant implications in various areas of mathematics and physics, including representation theory, quantum mechanics, and the study of differential equations. It allows for the translation of problems involving Lie algebras into problems involving associative algebras, which are often more tractable. The theorem is crucial for understanding the relationship between Lie groups and their Lie algebras, and it plays a role in the classification of Lie algebras.