Um Salama Ahmed Abd Alla Alemam* and Rasha Ahmed Hamid

Department of Mathematics, Al- Dayer University College, Jazan University, Kingdom of Saudi Arabia

*Corresponding Author: Um Salama Ahmed Abd Alla Alemam, Department of Mathematics, Al- Dayer University College, Jazan University, Kingdom of Saudi Arabia.

Received: October 10, 2023; Published: October 19, 2023

Abstract

In this study we have introduced Riemannian Manifold, Lie groups, Lie algebras, and root systems which help to give further understanding of symmetric spaces and some of their algebraic and topological properties which help in classification and many applications of symmetric spaces. I address and explore the basic concept of a root system. First, its origins in the theory of Lie algebras are introduced and then an axiomatic definition is provided. Bases, Weyl groups, and the transitive action of the latter on the former are explained Finally, the Cartan matrix and Dynkin diagram are introduced to suggest the multiple applications of root systems to other fields of study and their classification.

Keywords: Lie Groups; Lie Algebras; Topological Spaces; Metric Spaces; Topological Manifold; Riemannian Manifold; Root Systems; Symmetric Spaces; Weyl group; CatanMatrix and Dynkin Diagrams

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Citation

Citation: Um Salama Ahmed Abd Alla Alemam and Rasha Ahmed Hamid. “Root Systems, Cartan Matrix, Dynkin Diagrams in Classification of Symmetric Spaces"Acta Scientific Computer Sciences 5.11 (2023): 03-10.

Copyright

Copyright: © 2023 Um Salama Ahmed Abd Alla Alemam and Rasha Ahmed Hamid. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.