View source for Poincare-Birkhoff-Witt theorem

Given a [[Lie algebra]] <math>\mathfrak{g}</math> and an [[ordered basis]] of it , the '''Poincare-Birhkoff-Witt theorem''' constructs a [[basis (linear algebra)|basis]] for its [[universal envelopping algebra]] <math>U(\mathfrak{g})</math>, 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 <math>U(\mathfrak{g})</math>; And, with a polarisation of <math>\mathfrak{g}</math>, also a [[tensor product]] decomposition of <math>U(\mathfrak{g})</math>.

==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]]


[[category:Lie algebra]]
[[category:representation theory]]
[[category:mathematical theorems]]
[[category:algebra]]