Introduction

This papers deals with root systems R in a real vector space X which are defined in

analogy to the usual finite root systems a la Bourbaki [12, VI], except that finiteness

is replaced by local finiteness: The intersection of R with every finite-dimensional

subspace of X is finite.

Our aim is to develop the basic theory of these locally finite root systems. The

main topics of our work are Weyl groups, parabolic subsets and positive systems,

weights, and gradings. The reader will find that much, but not all, of the well-known

theory of finite root systems does generalize to this setting, although often different

proofs are needed. But there are also completely new phenomena, unfamiliar from

the theory of finite root systems. Most important among these is that a locally

finite root system R does in general not have a root basis, i.e., a vector space basis

B C R of X such that every root in R is an integer linear combination of B with

coefficients of the same sign. Thus, by necessity, our work presents a "basis-free"

approach to root systems. An important new tool is the concept of quotients of root

systems by full subsystems. When working with quotients, the usual requirement

that 0 ^ R proves to be cumbersome, so our root systems always contain 0. This

is also useful when considering root gradings of Lie algebras, and fits in well with

the axioms for extended affine root systems in [1, Ch. II]. It also occurs naturally

in the axiomatizations of root systems given by Winter [75] and Cuenca [19].

Throughout, we have attempted to develop the categorical aspect of root sys-

tems which, we feel, has hitherto been neglected. Thus we define the category RS

whose objects are locally finite root systems, and whose morphisms are linear maps

of the underlying vector spaces mapping roots to roots. Morphisms of this type

were studied for example by Dokovic and Tharig [25]. A more restricted class of

morphisms, called embeddings and defined by the condition that / preserve Cartan

numbers, leads to the subcategory RSE of RS whose morphisms are embeddings.

Many natural constructions, for example the coroot system, the Weyl group and

the group of weights, turn out to be functors defined on this category.

Let us stress once more that a locally finite root system is infinite if and only

if it spans an infinite-dimensional space. Hence, locally finite root systems are not

the same as the root systems appearing in the theory of Kac-Moody algebras. The

axiomatic approach to these types of root systems has been pioneered by Moody and

his collaborators [45, 48, 46]. Further generalizations are given in papers by Bardy

[4], Bliss [6], and Hee [30]. Roughly speaking, the intersection of locally finite root

systems and the root systems of Kac-Moody algebras consists of the direct sums

of finite roots systems and their countably infinite analogues, see Kac [35, 7.11]

or Moody-Pianzola [47, 5.8]. Similarly, the infinite root systems considered here

are not the same as the extended affine root systems which appear in the theory

of extended affine Lie algebras [1, Ch. II] and elliptic Lie algebras [66, 67]. The

extended affine root systems which are also locally finite root systems, are exactly

the finite root systems. Since extended affine root systems map onto finite root

systems, one is led to speculate that there should be a theory of "extended affine

l