ar X iv : m at h / 03 01 08 6 v 1 [ m at h . A G ] 9 J an 2 00 3 Maximal rank root subsystems of hyperbolic root systems

A Kac-Moody algebra is called hyperbolic if it corresponds to a generalized Cartan matrix of hyperbolic type. We study root subsystems of root systems of hyperbolic algebras. In this paper, we classify maximal rank regular hyperbolic subalgebras of hyperbolic Kac-Moody algebras. Introduction A generalized Cartan matrix A is called a matrix of hyperbolic type if it is indecomposable symmetrizable of indefinite type, and if any proper principal submatrix of the corresponding symmetric matrix B is of finite or affine type. In this case B is of the signature (n, 1). Consider a generalized Cartan matrix A of hyperbolic type. Following Kac [5], we can construct a Kac-Moody algebra g(A). According to Vinberg [8], the Weyl group of the root system ∆(A) is a Coxeter group. A fundamental chamber of the Weyl group is an n-dimensional hyperbolic Coxeter simplex of finite volume, whose dihedral angles are in the set {π2 , π 3 , π 4 , π 6 } (zero angle can also appear if n = 2). In analogy with the finite-dimensional theory (see [1]), we say a subalgebra g1 ⊂ g(A) to be regular if g1 is invariant with respect to some Cartan subalgebra h of g(A). In other words, g1 ⊂ g(A) is regular if it has a basis composed of some elements of h and some root vectors of g(A) (with respect to h). We are interested in maximal rank regular subalgebras that can be constructed as KacMoody algebras g1(A1) for some generalized Cartan matrix A1 of hyperbolic type. Any subalgebra of this type of the Kac-Moody algebra g(A) has a root system ∆1(A1) ⊂ ∆(A) such that (∗) if α, β ∈ ∆1 and α+ β ∈ ∆, then α+ β ∈ ∆1 Conversely, suppose we have a hyperbolic root system ∆1 in a hyperbolic root system ∆(A), and (∗) holds. Then we can construct a subalgebra of g(A) we are interested in. By hyperbolic root system we mean a root system of a Kac-Moody algebra constructed on a generalized Cartan matrix of hyperbolic type. Let ∆ be a hyperbolic root system. A root system ∆1 ⊂ ∆ is called a root subsystem of ∆ if the condition (∗) holds. The classification of root subsystems of finite root systems is due to Dynkin[1]. In this paper we classify maximal rank hyperbolic root subsystems of hyperbolic root systems. 1 Consider a maximal rank hyperbolic root subsystem ∆1 of a hyperbolic root system ∆. Let W1 and W be the Weyl groups of ∆1 and ∆ respectively. Let F1 and F be fundamental chambers of W1 and W . Then F1 and F are hyperbolic Coxeter simplices of finite volume. The groups W1 and W are generated by the reflections with respect to the facets of F1 and F respectively. Since W1 is a subgroup of W , the simplex F1 is composed of several copies of F . Moreover, any two copies of F having a common facet are symmetric with respect to this facet. By reflection group we mean a group generated by reflections. Introduce a partial ordering ≥ on the set of reflection subgroups of W by setting G ≥ H if H ⊂ G. A decomposition (F, F1) of a simplex F1 into several copies of F is called minimal if W1 is a maximal proper reflection subgroup of W . All the minimal decompositions of hyperbolic Coxeter simplices of finite volume are listed in [2], [3] and [6]. From now on by simplex we mean a hyperbolic Coxeter simplex of finite volume, whose dihedral angles are in the set {π2 , π 3 , π 4 , π 6 } (zero angle can also appear if n = 2). In Section 1 (Th. 1) we prove that any minimal decomposition of a hyperbolic simplex corresponds to some root subsystem of a hyperbolic root system. In Section 2 (Th. 2) we prove that any decomposition of a hyperbolic simplex corresponds to some root subsystem. The complete classification of maximal rank hyperbolic root subsystems is contained in Fig. 1–19. The author is grateful to Prof. E. B. Vinberg for his attention to the work and useful remarks. 1 Maximal subgroups We use the following notation: A is a generalized Cartan matrix of hyperbolic type; ∆ is the corresponding root system; α1, ..., αn+1 are simple roots; F is a fundamental chamber of ∆; L = n+1 ∑ i=1 Zαi is the corresponding root lattice. The Weyl group WF of ∆ is generated by the reflections with respect to the facets of the simplex F . The simple roots vanish on the facets of F . Furthermore, ∆ and L are the root system and the root lattice for the generalized Cartan matrix A (the fundamental simplex of the Weyl group of ∆ is the same as of ∆, but the lengths of simple roots are different in these systems). ∆1 ⊂ ∆ is a hyperbolic root system whose root lattice L1 is a maximal rank sublattice of L; F1 is a fundamental simplex of the Weyl group WF1 of the root system ∆1. We will use the following description of root system (see [5]). A hyperbolic root system ∆ consists of two disjoint parts: the set of real roots ∆ and the set of imaginary roots ∆, where ∆ = W (α1) ⋃ . . . ⋃ W (αn+1) , ∆ im = {α ∈ L | (α|α) ≤ 0}.