A Monstrous Lie group
Mathematical Physics and Probability Seminar
A Monstrous Lie group
Series: Mathematical Physics and Probability Seminar
Location: MATH 402
Presenter: Lisa Carbone, Rutgers University
The Monster Lie algebra m is a quotient of the physical space of the vertex algebra V=V^# \otimes V_{1,1}, where V^# is the Moonshine module of Frenkel, Lepowsky, and Meurman, and V_{1,1} is the vertex algebra corresponding to the rank 2 even unimodular lattice {II}_{1,1}. We discuss the construction of a Lie group analog G(m) associated to m. The group G(m) has a subgroup GL_2(-1) corresponding to the unique real simple root (1,-1) and an infinite family of subgroups GL_2(n,u,v) corresponding to imaginary simple roots (1,n) and suitable pairs of primary vectors u,v\in V^#. The Monster finite simple group M acts on this set of subgroups by taking GL_2(n,u,v) to GL_2(n,gu,gv) and fixing GL_2(-1). The group G(m) also has a subgroup U^+ analogous to a unipotent group, which acts on a completion \hat{m}=n^-\oplus h\oplus\hat{n}^+ of m. The group M acts on U^+ and the map exp ad :\hat{n}^+ --> U^+ is M-equivariant.
(zoom link: https://arizona.zoom.us/j/81196695512)