Symmetric Systems and their Applications to Root Systems Extended by Abelian Groups

Details Symmetric Systems and their Applications to Root Systems Extended by Abelian Groups Symmetric Systems and their Applications to Root Systems Extended by Abelian Groups

2007-12-02

arxiv.org/abs/0712.0104v2

Mathematics

Group Theory

38 pages

Scientific paper

We investigate the class of root systems R obtained by extending an irreducible root system by a torsion-free group G. In this context there is a Weyl group W and a group U with the presentation by conjugation. We show under additional hypotheses that the kernel of the natural homomorphism from U to W is isomorphic to the kernel of the homomorphism from the abelianization of U to that of W. For this we introduce the concept of a symmetric system, a discrete version of the concept of a symmetric space. Mathematics Subject Classification 2000: 20F55, 17B65, 17B67, 22E65, 22E40. Key Words and Phrases: Weyl group, root system, presentation by conjugation, extended affine Weyl group (EAWeG), extended affine root system (EARS), irreducible root system extended by an abelian group.

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