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Lie Groups pp 281-301 | Cite as

Relative Root Systems

  • Daniel Bump
Chapter
Part of the Graduate Texts in Mathematics book series (GTM, volume 225)

Abstract

In this chapter, we will consider root systems and Weyl groups associated with a Lie group G. We will assume that G satisfies the assumptions in Hypothesis 28.1 of the last chapter. Thus, G is semisimple and comes with a compact dual G c . In Chap.  18, we associated with G c a root system and Weyl group. That root system and Weyl group we will call the absolute root system Φ and Weyl group W. We will introduce another root system Φ rel, called the relative or restricted root system, and a Weyl group W rel called the relative Weyl group. The relation between the two root systems will be discussed. The structures that we will find give Iwasawa and Bruhat decompositions in this context. This chapter may be skipped with no loss of continuity.

Keywords

Root System Weyl Group Simple Root Short Exact Sequence Dynkin Diagram 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Daniel Bump
    • 1
  1. 1.Department of MathematicsStanford UniversityStanfordUSA

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