A lecture on Kac--Moody Lie algebras of the arithmetic type

TL;DR

This paper introduces the concept of arithmetic type for Kac--Moody Lie algebras, classifies these matrices into four types, and explores their connection with arithmetic groups generated by reflections in hyperbolic spaces.

Contribution

It generalizes the notion of hyperbolic type for Kac--Moody algebras and classifies matrices of arithmetic type into finite, affine, rank two, and hyperbolic categories.

Findings

Arithmetic type matrices are classified into four categories.

The hyperbolic type is closely related to arithmetic groups over $Q$.

Finite series of matrices of the hyperbolic type are described, with all symmetric cases known.

Abstract

We name an indecomposable symmetrizable generalized Cartan matrix $A$ and the corresponding Kac--Moody Lie algebra ${\goth g} ^\prime (A)$ {\it of the arithmetic type} if for any $\beta \in Q$ with $(\beta | \beta)<0$ there exist $n(\beta )\in {\Bbb N}$ and an imaginary root $\alpha \in \Delta^{im}$ such that $n(\beta )\beta \equiv \alpha \mod Ker\ (.|.)$ on $Q$. Here $Q$ is the root lattice. This generalizes "symmetrizable hyperbolic" type of Kac and Moody. We show that generalized Cartan matrices of the arithmetic type are divided in $4$ types: (a) finite, (b) affine, (c) rank two, and (d) arithmetic hyperbolic type. The last type is very closely related with arithmetic groups generated by reflections in hyperbolic spaces with the field of definition $\Bbb Q$. We apply results of the author and \'E.B. Vinberg on arithmetic groups generated by reflections in hyperbolic spaces to describe generalized Cartan matrices of the arithmetic hyperbolic type and to show that there exists a finite set of series of the generalized Cartan matrices of the arithmetic hyperbolic type. For the symmetric case all these series are known.