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

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.

• 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