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Friday, 12/02/2016

posted Feb 13, 2016, 11:08 PM by David Radnell   [ updated Feb 13, 2016, 11:51 PM ]
We continued with algebraic basics following Lepowsky and Li (pages 57 - 70, 84). In particular we discussed
  • Commutator and Associator forumlas
  • Weak commutativity, weak associativity and the D-derivative property. We sketched the proof of weak commutativity and outlined two of the key ingredients in proving the D-derivative property. Sugested HW: Fill in the details of these proofs.
  • Stated the theorem that in the definition of a vertex algebra, the Jacobi Identity can be replaced with "weak commutativity + D-derivative property".
  • Gave the definition of a VOA and discussed some basic properties of the conformal vector \omega and the formulas involving L(0), L(-1) and L(-2). 
  • Very briefly discussed the restricted dual, "rationality of products" and the "commutativity" in the sense that of different expansions of a common rational function. This discussion was rushed at the end and I may have said something incorrect. Please see page 68, expression (3.2.4) and Proposition 3.2.7. Suggested HW: Look at the proof and fill in the details of this conceptually important proposition. 
  • Highly recommended general reading: Terry Gannon, Vertex Operator Algebras. In the Princeton Guide to Mathematics. (A nice short 11 page overview of VOA)