Acta Scientific Computer Sciences

Review Article Volume 5 Issue 11

Root Systems, Cartan Matrix, Dynkin Diagrams in Classification of Symmetric Spaces

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

References

  1. Dyson F. Communications in Mathematical Physics 19 (1970): 235.
  2. Olshanetsky MA and Perelomov AM. Physical Report 94 (1983): 313.
  3. Hermann R. “Lie Groups for physicists”. (W.A Benjamin Inc. , New York) (1966).
  4. Ulrika Magnea. “An introduction to symmetric spaces”. Dept. Of Math university of Torino 10,1-10125, Italy.
  5. Stephanie Kernik. “A very brief overview of lie algebras”. University of Minnesota, Morris, (2008).
  6. Humphreys James E. “Introduction to Lie algebras and representation theory”. Springer, (1994).
  7. Sattinger DH and weaver OL. “Lie Group and Lie algebras with application to physics, geometry and mechanics”. Springer – verlag, new York (1986).
  8. Boothby WM. “An Introduction to differentiable Manifolds and Riemannian Geometry”. Academic Press, New York (1975).
  9. Helgason S. “Differential geometry, Lie Groups and symmetric spaces”. Academic press, New York (1978).
  10. Gilmore R. “Lie Groups, lie algebras and Some of Their Applications”. John wiley and Sons, New York (1974).
  11. M Brion. “Representations of Quivers”. Unpublished lecture notes (2008).
  12. H Derksen and J Weyman. “Quiver Representations”. Notices of the AMS 52.2 (2005).
  13. P Etingof., et al. “Introduction to Representation Theory”. Unpublished lecture notes. (2009).
  14. J E Humphreys. “Introduction to Lie Algebras and Representation Theory”. Springer-Verlag. 1972 Theory. Springer. (1983).
  15. V G Kac. “Root Systems, Representations of Quivers and Invariant Theory”. Springer (1983).

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.




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Acceptance rate35%
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