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.
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978-0-8218-3546-3 (9780821835463)
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Schweitzer Klassifikation
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.
