Advances in Linear Algebra & Matrix Theory, 2023, 13, 1-20 https://www.scirp.org/journal/alamt ISSN Online: 2165-3348 ISSN Print: 2165-333X DOI: 10.4236/alamt.2023.131001 Mar. 31, 2023 1 Advances in Linear Algebra & Matrix Theory Root Systems Lie Algebras Amor Hasić 1 , Fatih Destović 2 1 Department of Computer Science, International University of Novi Pazar, Novi Pazar, Serbia 2 Faculty of Educational Sciences, University of Sarajevo, Sarajevo, Bosnia and Herzegovina Abstract A root system is any collection of vectors that has properties that satisfy the roots of a semi simple Lie algebra. If g is semi simple, then the root system A, (Q) can be described as a system of vectors in a Euclidean vector space that possesses some remarkable symmetries and completely defines the Lie algebra of g. The purpose of this paper is to show the essentiality of the root system on the Lie algebra. In addition, the paper will mention the connection be- tween the root system and Ways chambers. In addition, we will show Dynkin diagrams, which are an integral part of the root system. Keywords Lie Algebras, Root Systems, Ways Chambers, Dynkin Diagrams 1. Introduction A root system in mathematics is a configuration of vectors in Euclidean space that satisfies certain geometric properties. The concept is fundamental in the theory of Lie algebras and Lie groups, especially in the theory of classification and representation of semisimple Lie algebras. Since Lie groups (and some analogues such as algebraic groups) and Lie algebras became important in many parts of mathematics during the twentieth century, the apparently special nature of root systems is inconsistent with the number of areas in which they are applied. Definition 1.1. A root system ( ) , VR is a finite-dimensional real vector space V wich an inner product (i.e. a Euclidean vector space), such that the following properties hold: a) The vectors in R span V. b) If α is in R and c R ∈ , then cα is in R only if 1 c =± , then R is called a reduced root system. How to cite this paper: Hasić, A. and Destović, F. (2023) Root Systems Lie Alge- bras. Advances in Linear Algebra & Matrix Theory, 13, 1-20. https://doi.org/10.4236/alamt.2023.131001 Received: October 7, 2023 Accepted: March 28, 2023 Published: March 31, 2023 Copyright © 2023 by author(s) and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ Open Access