# Poincare-Birkhoff-Witt theorem

Given a Lie algebra and an ordered basis of it , the **Poincare-Birhkoff-Witt theorem** constructs a basis for its universal envelopping algebra , called the **Poincare-Birkhoff-Witt** (**PBW** for short) **basis**, consisting of the lexographically ordered monomials of the basis elements. This theorem is fundamental in representation theory. It gives an concrete description of ; And, with a polarisation of , also a tensor product decomposition of .

## Exercise Edit

Write out the PBW basis for sl2.

## References Edit

- Frenkel, ben-Zvi,
*Vertex algebras and algebraic curves*, p.27 (brief) - James. E. Humphreys,
*Introduction to Lie algebras and representation theory*, pp.91-93 (detailed)