Abstract:

A vertex operator algebra $V=V_0+V_1+...$ contains infinitely many nonassociative finite-dimensional algebras $V_k$.
$V_0$ is just the field, and $V_1$ is a Lie algebra. Of course this Lie algebra plays an important role, organizing
the structure of the full vertex operator algebra V. Unfortunately, in many important examples (e.g. the Monstrous
Moonshine VOA), $V_1=0$. When $V_1=0$, what kind of algebra is $V_2$, and how does it organize V? I'll discuss this
in my talk. I won't assume any knowledge of vertex algebras, though some familiarity with Jordan algebras wouldn't hurt.