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Published in:
2017 | OriginalPaper | Chapter
3. Root Systems
Author : Xiaoping Xu
Published in: Representations of Lie Algebras and Partial Differential Equations
Publisher: Springer Singapore
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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.