Poincare-Birkhoff-Witt theorem

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

Exercise
Write out the PBW basis for sl2.

Wikimedia resources

 * PBW theorem