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Locally Finite Root Systems
Ottmar Loos, University of Innsbruck, Austria, and Erhard Neher, University of Ottawa, ON, Canada

Memoirs of the American Mathematical Society
2004; 214 pp; softcover
Volume: 171
ISBN-10: 0-8218-3546-7
ISBN-13: 978-0-8218-3546-3
List Price: US$60
Individual Members: US$36
Institutional Members: US$48
Order Code: MEMO/171/811
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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 finite-dimensional subspace of \(X\) is finite. The main topics are Weyl groups, parabolic subsets and positive systems, weights, and gradings.


Graduate students and research mathematicians interested in infinite-dimensional 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 3-graded root systems
  • Some standard results on finite root systems
  • Cones defined by totally preordered sets
  • Bibliography
  • Index of notations
  • Index
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