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 | edit source]
Write out the PBW basis for sl2.
References[edit | edit source]
- 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)