A Lie group analog for the Monster Lie algebra

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~$G(\mathfrak{m})$ of a Lie group or Kac–Moody group, associated to~$\mathfrak{m}$. The group~$G(\mathfrak{m})$ 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 $\mathfrak{m}$, we introduce the notion of pro-summability of an infinite sum of operators. We use this to construct a complete pro-unipotent group $\text{Uhp}$ of automorphisms of a completion $\widehat{\mathfrak{m}}={\mathfrak{n}}^{-}\oplus \mathfrak{h}\oplus {\widehat{\mathfrak{n}}}^{+}$ of~$\mathfrak{m}$, where ${\widehat{\mathfrak{n}}}^{+}$ is the formal product of the positive root spaces of $\mathfrak{m}$. 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~$\mathfrak{m}$.We construct analogs $\text{Exp}:{\widehat{\mathfrak{n}}}^{+}\to {\widehat{U}}^{+}$ and $\text{Ad}:{\widehat{U}}^{+}\to \text{Aut}({\widehat{\mathfrak{n}}}^{+})$ of the classical exponential map and adjoint representation. We show that the action of $\mathbb{M}$ on $\mathfrak{m}$ induces an action of~$\mathbb{M}$ on~$\widehat{\mathfrak{m}}$, and that this in turn induces an action of $\mathbb{M}$ on~${\widehat{U}}^{+}$. We also show that the action of $\mathbb{M}$ on ${\widehat{\mathfrak{n}}}^{+}$ is compatible with the action of ${\widehat{U}}^{+}$ on ${\widehat{\mathfrak{n}}}^{+}$.