Poincare-Birkhoff-Witt theorem

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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)

Wikimedia resources[edit]