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

2007/12/02 by Georg Hofmann, Georg W. Hofmann, Hofmann, Georg W.
Mathematics · #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Group Theory (math.GR) #math.GR

paper · pdf · doi:10.48550/arxiv.0712.0104

38 pages

openalex publication_date 2007/12/02 · arxiv created 2007/12/05 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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.

Related

• The Weyl group of type A₁ root systems extended by an abelian group
• System of roots
• Presentation by conjugation for A₁-type extended affine Weyl groups
• A classification of subsystems of a root system
• A characterization of minimal extended affine root systems (Relations to Elliptic Lie Algebras)