The Poincare-Birkhoff-Witt Theorem describes the structure of the universal enveloping algebra of a Lie algebra, showing that it has a vector-space basis given by ordered products of a basis of the Lie algebra, exactly as a naive symmetrization of the Lie algebra's underlying vector space would suggest. Named for Henri Poincare, Garrett Birkhoff and Ernst Witt, it is a foundational result linking Lie algebras to associative algebra and underlies much of representation theory.
Facts
Partially Attested
Proof YearPoincare stated a general form in 1900; Birkhoff and Witt independently proved it in 1937, so the year is for the first statement, not the accepted proof. Classification
Statement FormCharacterization Theorem 1 Connections
Has Statement Form
In Branch
Source Poincare-Birkhoff-Witt theorem (Wikipedia)
Proved By
Sources
1. Poincare-Birkhoff-Witt theorem (Wikipedia)
History of the theorem section
Poincaré later stated it more generally in 1900.
- In Branch: Lie Theory, Lead sentence
In Group: Lie Algebras and Representation Theory, Lead section
In mathematics, more specifically in the theory of Lie algebras, the Poincaré-Birkhoff-Witt theorem (or PBW theorem)
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