2023/11/18 by Lisa Carbone, Elizabeth Jurisich, Carbone, Lisa +3
Mathematics · #Algebraic structures and combinatorial models #Advanced Algebra and Geometry #Finite Group Theory Research
paper · pdf · doi:10.48550/arxiv.2311.11078
The Monster Lie algebra \frak m , which admits an action of the Monster finite simple group \mathbbM, was introduced by Borcherds as part of his work on the Conway--Norton Monstrous Moonshine conjecture. Here we construct an analog~G(\frak m) of a Lie group or Kac--Moody group, associated to~\frak m. The group~G(\frak m) is given by generators and relations, analogous to a construction of a Kac--Moody group given by Tits. In the absence of local nilpotence of the adjoint representation of \frak m, we introduce the notion of pro-summability of an infinite sum of operators. We use this to construct a complete pro-unipotent group \Uhp of automorphisms of a completion \widehat\mathfrakm=\frak n- ⊕ \frak h ⊕ \widehat\frak n+ of~\mathfrakm, where \widehat\frak n+ is the formal product of the positive root spaces of \frak m. The elements of \widehatU+ are pro-summable infinite series with constant term 1. The group \widehatU+ has a subgroup~\widehatU+im, which is an analog of a complete unipotent group corresponding to the positive imaginary roots of~\frak m.We construct analogs Exp: \widehat\mathfrakn+→\widehatU+ and Ad :\widehatU+ → \Aut(\widehat\frakn+) of the classical exponential map and adjoint representation. We show that the action of \mathbbM on \mathfrak m induces an action of~\mathbbM on~\widehat\frak m, and that this in turn induces an action of \mathbbM on~\widehatU+. We also show that the action of \mathbbM on \widehat\mathfrak n+ is compatible with the action of \widehatU+ on \widehat\mathfrak n+.