Title: A Lie group analog for the Monster Lie algebra

Abstract: The Monster Lie algebra $\mathfrak{m}$, which admits an action of the Monster finite simple group $\mathbb{M}$, 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 the Tits construction of a Kac-Moody group. 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 $\widehat{U}^+$ of automorphisms of a completion $\widehat{\mathfrak{m}}=\frak n^-\ \oplus\ \frak h\ \oplus\ \widehat{\frak n}^+$ of $\mathfrak{m}$, where $\widehat{\frak n}^+$ is the formal product of the positive root spaces of $\frak m$. The elements of $\widehat{U}^+$ are pro-summable infinite series with constant term 1. The group $\widehat{U}^+$ has a subgroup $\widehat{U}^+_{\text{im}}$, which is an analog of a complete unipotent group corresponding to the positive imaginary roots of $\frak m$. We construct analogs Exp:$\widehat{\mathfrak{n}}^+\to\widehat{U}^+$ and Ad:$\widehat{U}^+ \to Aut(\widehat{\frak{n}}^+)$ of the classical exponential map and adjoint representation. Although the group $G(\mathfrak m)$ is not a group of automorphisms, it contains the analog of a unipotent subgroup $U^+$, which conjecturally acts as automorphisms of $\widehat{\mathfrak{m}}$.
We also construct groups of automorphisms of $\mathfrak{m}$, of certain $\mathfrak{gl}_2$ subalgebras of $\mathfrak{m}$, of the completion $\widehat{\mathfrak{m}}$ and of similar completions of $\frak m$ that are conjecturally identified with subgroups of $G(\mathfrak m)$.