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Root Systems

In: Representations of Lie Algebras and Partial Differential Equations

Author

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  • Xiaoping Xu

    (Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Institute of Mathematics
    University of Chinese Academy of Sciences, School of Mathematics)

Abstract

We start with the axiom of root system and give the root systems of special linear algebras, orthogonal Lie algebras and symplectic Lie algebras. Then we derive some basic properties of root systems. As finite symmetries of root systems, the Weyl groups are introduced and studied in detail. The classification and explicit constructions of root systems are presented. The automorphism groups of root systems are determined. As a preparation for later representation theory of Lie algebras, the corresponding weight lattices and their saturated subsets are investigated.

Suggested Citation

  • Xiaoping Xu, 2017. "Root Systems," Springer Books, in: Representations of Lie Algebras and Partial Differential Equations, chapter 0, pages 61-93, Springer.
    • Handle: RePEc:spr:sprchp:978-981-10-6391-6_3
    DOI: 10.1007/978-981-10-6391-6_3
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