A Lecture on Kac–moody Lie Algebras of the Arithmetic Type

We name an indecomposable symmetrizable generalized Cartan matrix A and the corresponding Kac–Moody Lie algebra g(A) of the arithmetic type if for any β ∈ Q with (β|β) < 0 there exist n(β) ∈ N and an imaginary root α ∈ ∆ such that n(β)β ≡ α 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 Q. We apply results of the author and É.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. 0. Introduction. This lecture was given by the author at Johns Hopkins University, Notre Dame University, Penn State University and Queen’s University on the fall 1994. I am grateful to these Universities for hospitality. I am grateful to Professor V. Chari for useful discussions. I am grateful to Professor É.B. Vinberg for his interest to this subject. We want to pay attention to one class of Kac–Moody Lie algebras which is very closely related with arithmetic reflection groups in hyperbolic spaces. 1. Reminding on symmetrizable Kac–Moody Lie algebras. Here we recall results on symmetrizable Kac–Moody Lie algebras which we need. One can find them in the book by Victor Kac [Ka1]. (1.1) An n× n-matrix A = (aij) is called a generalized Cartan matrix if (C1) aii = 2 for i = 1, ..., n; (C2) aij are non-positive integers for i 6= j; (C3) aij = 0 implies aji = 0. We denote by l the rank of A and by k = n− l the dimension of the kernel of A. For simplicity, below we suppose that A is indecomposable which means that there does not exist a decomposition I = {1, ..., n} = I1 ∪ I2 such that both I1 and I2 are non-empty and aij = 0 for any i ∈ I1 and any j ∈ I2. Partially supported by Grant of Russian Fund of Fundamental Research; Grant of AMS; and Grant of ISF MI6000.