Memoirs of the American Mathematical Society 2004; 214 pp; softcover Volume: 171 ISBN10: 0821835467 ISBN13: 9780821835463 List Price: US$60 Individual Members: US$36 Institutional Members: US$48 Order Code: MEMO/171/811
 We develop the basic theory of root systems \(R\) in a real vector space \(X\) which are defined in analogy to the usual finite root systems, except that finiteness is replaced by local finiteness: The intersection of \(R\) with every finitedimensional subspace of \(X\) is finite. The main topics are Weyl groups, parabolic subsets and positive systems, weights, and gradings. Readership Graduate students and research mathematicians interested in infinitedimensional Lie theory. Table of Contents  Introduction
 The category of sets in vector spaces
 Finiteness conditions and bases
 Locally finite root systems
 Invariant inner products and the coroot system
 Weyl groups
 Integral bases, root bases and Dynkin diagrams
 Weights and coweights
 Classification
 More on Weyl groups and automorphism groups
 Parabolic subsets and positive systems for symmetric sets in vector spaces
 Parabolic subsets of root systems and presentations of the root lattice and the Weyl group
 Closed and full subsystems of finite and infinite classical root systems
 Parabolic subsets of root systems: classification
 Positive systems in root systems
 Positive linear forms and facets
 Dominant and fundamental weights
 Gradings of root systems
 Elementary relations and graphs in 3graded root systems
 Some standard results on finite root systems
 Cones defined by totally preordered sets
 Bibliography
 Index of notations
 Index
