[Paper Review] Infinite dimensional Chevalley groups and Kac-Moody groups over $\mathbb{Z}$

Lisa Carbone, Dongwen Liu+1 more|arXiv (Cornell University)|Mar 29, 2018

Algebraic structures and combinatorial models23 references3 citations

TL;DR

This paper constructs infinite-dimensional Chevalley-type groups $G_V(\mathbb{Z})$ from $\mathbb{Z}$-forms of Kac-Moody algebras and integrable highest-weight modules over $\mathbb{Z}$, generalizing arithmetic subgroups to non-affine, symmetrizable Kac-Moody groups. It establishes a Bruhat decomposition for the completion $\widetilde{G}(\mathbb{Q})$ and proves that the arithmetic subgroup $\widetilde{\Gamma}(\mathbb{Z})$ equals the group of integral points $\widetilde{G}(\mathbb{Z})$.

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 $\widetilde{G}$. For the completion we prove a Bruhat decomposition $\widetilde{G}({\mathbb{Q}})=\widetilde{G}({\mathbb{Z}})\widetilde{B}({\mathbb{Q}})$ over $\mathbb{Q}$, and that the arithmetic subgroup $\widetildeΓ(\mathbb{Z})$ coincides with the subgroup of integral points $\widetilde{G}(\mathbb{Z})$

Motivation & Objective

• Develop a construction of infinite-dimensional analogues of Chevalley groups over $\mathbb{Z}$ for non-finite, non-affine Kac-Moody algebras.
• Define arithmetic subgroups $G_V(\mathbb{Z})$ using $\mathbb{Z}$-forms of universal enveloping algebras and integrable highest-weight modules.
• Introduce a universal representation-theoretic Kac-Moody group $G$ and its completion $\widetilde{G}$ to study arithmetic structures.
• Establish a Bruhat decomposition for $\widetilde{G}(\mathbb{Q})$ in terms of $\widetilde{G}(\mathbb{Z})$ and $\widetilde{B}(\mathbb{Q})$.
• Prove that the arithmetic subgroup $\widetilde{\Gamma}(\mathbb{Z})$ coincides with the group of integral points $\widetilde{G}(\mathbb{Z})$.

Proposed method

• Use a symmetrizable generalized Cartan matrix $A$ not of finite or affine type to define the Kac-Moody algebra $\mathfrak{g}$ over a commutative ring $R$ with $1$.
• Construct a $\mathbb{Z}$-form of the universal enveloping algebra of $\mathfrak{g}$ to define integral structures.
• Utilize a $\mathbb{Z}$-form of an integrable highest-weight module $V$ to define the group $G_V(R)$ as a generalization of finite-dimensional Chevalley groups.
• Define the universal Kac-Moody group $G$ and its completion $\widetilde{G}$ to enable topological and arithmetic analysis over $\mathbb{Q}$.
• Apply representation-theoretic techniques to establish the Bruhat decomposition $\widetilde{G}(\mathbb{Q}) = \widetilde{G}(\mathbb{Z})\widetilde{B}(\mathbb{Q})$.
• Use the structure of the completion $\widetilde{G}$ to prove that $\widetilde{\Gamma}(\mathbb{Z}) = \widetilde{G}(\mathbb{Z})$, identifying the arithmetic subgroup with the integral points.

Experimental results

Research questions

• RQ1 How can one construct infinite-dimensional analogues of Chevalley groups over $\mathbb{Z}$ for non-finite, non-affine Kac-Moody algebras?
• RQ2 What is the role of $\mathbb{Z}$-forms of the universal enveloping algebra and integrable highest-weight modules in defining arithmetic subgroups?
• RQ3 Does a Bruhat decomposition hold for the completion $\widetilde{G}(\mathbb{Q})$ of the universal Kac-Moody group?
• RQ4 Is the arithmetic subgroup $\widetilde{\Gamma}(\mathbb{Z})$ equal to the group of integral points $\widetilde{G}(\mathbb{Z})$ in the completed group?
• RQ5 How do representation-theoretic constructions extend the theory of arithmetic groups to infinite-dimensional Kac-Moody groups?

Key findings

• The paper constructs infinite-dimensional Chevalley-type groups $G_V(\mathbb{Z})$ using $\mathbb{Z}$-forms of the universal enveloping algebra and integrable highest-weight modules over $\mathbb{Z}$.
• A universal representation-theoretic Kac-Moody group $G$ and its completion $\widetilde{G}$ are defined to facilitate arithmetic and topological analysis.
• The completion $\widetilde{G}$ admits a Bruhat decomposition over $\mathbb{Q}$: $\widetilde{G}(\mathbb{Q}) = \widetilde{G}(\mathbb{Z})\widetilde{B}(\mathbb{Q})$.
• The arithmetic subgroup $\widetilde{\Gamma}(\mathbb{Z})$ is shown to coincide exactly with the group of integral points $\widetilde{G}(\mathbb{Z})$.
• The construction generalizes the classical notion of arithmetic subgroups to the infinite-dimensional setting of non-affine Kac-Moody groups.