Infinite dimensional Chevalley groups and Kac-Moody groups over $\mathbb{Z}$

Abstract

Let $A$ be a symmetrizable generalized Cartan matrix, which is not of finite or affine type. Let $\mathfrak{g}$ be the corresponding Kac-Moody algebra over a commutative ring $R$ with $1$. We construct an infinite-dimensional group $G_V(R)$ analogous to a finite-dimensional Chevalley group over $R$. We use a $\mathbb{Z}$-form of the universal enveloping algebra of $\mathfrak{g}$ and a $\mathbb{Z}$-form of an integrable highest-weight module $V$. We construct groups $G_V(\mathbb{Z})$ analogous to arithmetic subgroups in the finite-dimensional case. We also consider a universal representation-theoretic Kac-Moody group $G$ and its completion $\tilde{G}$. For the completion we prove a Bruhat decomposition $\tilde{G}(\mathbb{Q})=\tilde{G}(\mathbb{Z})\tilde{B}(\mathbb{Q})$ over $\mathbb{Q}$, and that the arithmetic subgroup $\tilde{\Gamma }(\mathbb{Z})$ coincides with the subgroup of integral points $\tilde{G}(\mathbb{Z})$

Proposition 6.4 fails for imaginary roots. This error propagates through other results and the main theorem cannot be proven without Proposition 6.4. A partial proof using a different method can be found here arXiv:2210.01644

References in corpus (1)

• Kac--Moody groups and automorphic forms in low dimensional supergravity theories

Cited by in corpus (1)

• Twin masures associated with Kac-Moody groups over Laurent polynomials