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Lie Algebra, Cartan Subalgebra, Root System, Symmetric Space American Journal of Mathematics and Statistics 2016, 6(5): 197-202 DOI: 10.5923/j.ajms.20160605.01 The Role of Root System in Classification of Symmetric Spaces M-Alamin A. H. Ahmed1,2 1Department of Mathematics, Faculty of Science and Arts – Khulais, University of Jeddah, Saudi Arabia 2Department of Mathematics, Faculty of Education, Alzaeim Alazhari University (AAU), Khartoum, Sudan Abstract In this paper we have introduced a thorough study of Lie algebra, disclosing its contribution to classification of symmetric spaces via root systems. Any Lie algebra is associated to its Lie group through the exponential mapping, and also the Lie algebra corresponds to a given root system which gives its classification. A symmetric space can be represented as a coset space and so we can introduce a symmetric space algebraically using Lie groups and their Lie algebras, then by introducing restricted root systems we can classify symmetric spaces. We gave the theoretical background of this classification with some examples which helps in understanding and further study of this topic. Keywords Lie algebra, Cartan subalgebra, Root system, Symmetric Space Root systems are also the key ingredient in the 1. Introduction classification of finite –dimensional, simple Lie algebras. ɡ Symmetric spaces are special topic in Riemannian Corresponding to a simple Lie algebra we have a geometry, they were earlier studied and classified by Elie Cartan decomposition and so we have a root system. Since a Cartan (1869- 1951), and since then many scholars studied symmetric space is a homogeneous space that can be them and gave many of their applications in mathematics, represented as a coset space by using Lie groups and their physics and other scientific fields [3]. Symmetric spaces can Lie algebras, so Lie algebras and their root systems play a be introduced through different approaches. For instance fundamental role in classification of symmetric spaces [3]. algebraically they can be introduced through Lie groups and This classification is a continuous field of scientific research. their Lie algebras or geometrically by using curvature tensor. So we aim at giving the tools for this classification in our They can be viewed as Riemannian manifolds with point current paper. reflections [2], or with parallel curvature tensor or as a homogeneous space with special isotropy group or a Lie group with a certain involution and so on. The Fundamental 2. Lie Algebra property of Lie theory is that one may associate with any Lie A Lie algebra ɡ is a vector space with skew – symmetric group G a Lie algebra ɡ [1] & [4] & [6]. The Lie algebra is a ɡ ɡ vector space with properties that make it possible to deal bilinear map, called Lie bracket and written as [.,.] : × with using tools of linear algebra. The Lie group G is almost → ɡ which satisfies the Jacobi identity completely determined by its Lie algebra ɡ. There is a basic ++ ɡ → G x,,[ yz] y ,,[ zx] z ,,[ xy] connection between the two structures given by exp: [5] & [7]. For many scientific problems, the complicated =∀∈0,xyz , , ɡ nonlinear structure of the Lie group can be reformulated using the exponential map in the Lie algebra, and this makes 2.1. The Lie Algebra of a Lie Group it easy to use tools of linear algebra especially when we use Cartan subalgebras. This is defined as the tangent space to the Lie group at the A Lie group also is a differentiable manifold, and this identity. make it possible to join symmetric spaces as differentiable Here are some examples of important Lie algebras: manifolds also. 2.1.1. Examples * Corresponding author: For a field k of characteristic zero, we have the classical [email protected] (M-Alamin A. H. Ahmed) matrix algebras gln ( k ) of nn× matrices over k, sln ( k ) Published online at http://journal.sapub.org/ajms Copyright © 2016 Scientific & Academic Publishing. All Rights Reserved the subalgebra of gln ( k ) of those nn× matrices with 198 M-Alamin A. H. Ahmed: The Role of Root System in Classification of Symmetric Spaces ii) If α∈Φ and λα∈Φ , then λ =± determinant one. There are also the algebras sOn () of 1 iii) If α∈Φ , then Φ is closed under reflection through nn× orthogonal real matrices, or sUn () of nn× the hyperplane normal to α . unitary complex matrices. The bracket operation for all these iv) If αβ, ∈Φ , then is given by [X, Y] = XY − YX . 2,(αβ) 2.2. Definition (Ideals, Simple and Semisimple Lie αβ,:= ∈ (3.1) (ββ, ) Algebras) α∈Φ An ideal I of a Lie algebra ɡ is a vector space of ɡ such that The elements are called roots. [a,, b]∈ I ∀∈ a I and b ∈ɡ If θ is the angle between αβ& , then the possible A simple Lie algebra is the one which has no proper ideal. values of θ are: 0,,,,,,π ππ23 ππ π , Also a semisimple Lie algebra is the one which is a direct 23 34 4 sum of simple Lie algebras. ππ, 5 . This can be shown using the relation 66 2 2.2.1. Example 4,(αβ) αβ, = =4cos2 θ nn× 22 αβ (3.2) Let sln () be the set of all matrices of trace 0. αβ⋅ sln () is an ideal of gln () which is nonzero. So 3.1. Examples gln () is not simple. (i) {±(eij − e)|1 ≤≤ i jn ≤ + 1} is the root system of the 2.3. Theorem [3] Lie algebra A= sL( n +1,) . For a matrix group G⊆ GL( n, V ) (general linear n (ii) The set of standard basis vectors and their opposites group), the set {±ei |1 ≤≤ in} is a root system. ɡ =∈∈{X EndV:exp( tA) } , is a Lie algebra, called the Lie algebra of G. 3.2. The Weyl Group [5] The symmetry of a root system defined by reflection 2.4. Proposition [4] through the hyperplane perpendicular to α is given by Let α :GH→ be a continuous homomorphism between σ( β)= β − αβα, (3.3) matrix groups. Then there exists a unique Lie algebra α homomorphism dhα :ɡ→ such that the following diagram The group generated by {σαα : ∈Φ} is the weyl group commutes: of the system. GHα → 3.3. Decomposable and Indecomposable Root System exp↑↑ exp A root system Φ is said to be decomposable if it can be dα ɡ → h written as Φ=Φ ∪Φ such that (αα,0)= for all 12 12 αα∈Φand ∈Φ . We say Φ is indecomposable if it is 2.5. Definition (The Adjoint Representation) 11 2 2 not decomposable. Let ɡ be a Lie algebra over k. A representation of ɡ is a Lie Every root system can be written as the disjoint union of algebra homomorphism ρ:,ɡ→ gl( n k ) for some n called indecomposable root systems. the degree of the representation. We define a mapping 3.4. Positive and Simple Roots ad X from a Lie algebra to itself by adX:, Y→[ X Y ]. A positive root is one such that its first non-zero element → The mapping X ad X is a representation of the Lie (in the chosen basis) is positive. algebra called the adjoint representation. It is an + If we denote the set of positive roots by Φ ⊆Φ, they automorphism. satisfy : (1) ∀α ∈Φ , exactly ααor − ∈Φ+ 3. Root System (2) If αβ, ∈Φ+ , and if αβ+ is a root, then A root system Φ is a set of vectors in n such that: + αβ+ ∈Φ . n i) Φ spans and 0∉Φ The negative roots are the nonpositive roots. A simple root American Journal of Mathematics and Statistics 2016, 6(5): 197-202 199 for Φ is α∈Φ if it is not the sum of two other positive n Where ee12, ,... en are an orthonormal basis of . The roots. We denote the set of simple roots by Π . dimension of the system is indicated by its subscript, so all To find a set of simple roots, we must determine firstly span n except A . Also we have the exceptional root when two roots may be added together. If the angle θ n−1 αβ∈Φ systems GFEE2467,,, & E 8. between the two roots , is strictly obtuse, then αβ+ ∈Φ , If θ is strictly acute and βα≥ then αβ− ∈Φ . 4. Root System and Cartan Subalgebra Suppose ɡ is a complex simple Lie algebra with a vector 3.5. Theorem [4] space basis {xx12, ,... xn} . With respect to this basis we can Every root system has a set of simple roots Π such that discuss the structure of the Lie algebra ɡ. So we find the each α ∈Φ may be written as a linear combination of structure constants fi jk such that αγ∈ k elements of Π , that is ∑ γ , with kγ ∈ , and n γ ∈Π {xxi, j}=∑ f i jk (4.1) each kγ has the same sign. k =1 If as many as possible these structure constants are zeros, 3.6. Lemma [4] then we can find most of the information of ɡ through the The set of simple roots Π is an independent set, and is a constants fi jk , s. So we find what is called Cartan basis for n . subalgebra. 3.7. Height of a Root 4.1. Cartan Subalgebra ɧ A cartan subalgebra ɧ for a Lie algebra ɡ is a subalgebra If we fix a base Π⊂Φ, Π={αα12, ,......, αn} so that satisfying the following conditions: α ,1≤≤in are simple roots, for a root αα= c we i ∑ ii i) ɧ is a maximal abelian subalgebra of ɡ. define htα = c as the height of α .
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