Given a Lie algebra ${\displaystyle {\mathfrak {g}}}$ and an ordered basis of it , the Poincare-Birhkoff-Witt theorem constructs a basis for its universal envelopping algebra ${\displaystyle U({\mathfrak {g}})}$, 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 ${\displaystyle U({\mathfrak {g}})}$; And, with a polarisation of ${\displaystyle {\mathfrak {g}}}$, also a tensor product decomposition of ${\displaystyle U({\mathfrak {g}})}$.

Exercise

Write out the PBW basis for sl2.

References

• 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)

Wikimedia resources

• w:PBW theorem