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
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