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American Journal of Mathematics and Statistics 2016, 6(5): 197-202 DOI: 10.5923/j.ajms.20160605.01

The Role of 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 , disclosing its contribution to classification of symmetric spaces via root systems. Any Lie algebra is associated to its through the exponential mapping, and also the Lie algebra corresponds to a given root system which gives its classification. A 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, , 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 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 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 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 [5] The 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. GHα → 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 ) 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 . 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 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 . ii) For each H ∈ ɧ, the endomorphism ad H of ɡ is ∑ i In an irreducible root system shorter roots are called short semisimple. and longer roots are called long. A cartan subalgebra is diagonalizable subalgebra which is maximal under set inclusion. Its dimension is the rank of ɡ. 3.8. Proposition [4] All Cartan subalgebras of a Lie algebra ɡ are conjugate under of ɡ, and they have the same Let Φ be an irreducible root system. Then at most two different root lengths occur in Φ . dimension. ɧ ɧ Define the basis {H1 ,...,Hr } for . Since is abelian, 3.9. Remark HH,0= for all ij, . Any root is an image of a simple root under the action of ij the weyl group. We extend this basis for ɧ to a basis for ɡ, and then we get a much simpler basis for ɡ with convenient commutator Using the closure of Φ under reflections σα , that is relations. elements of the weyl group, we can reconstruct the entire ɡ root system. The adjoint operators for Hi form a representation of , Any root is an image of a simple root under the action of called the adjoint representation. These operators adjH j the weyl group. have a set of common eigenvectors, and more over, by the ɡ 3.10. Classification of Root System and Lie Algebra spectral theorem we have decomposition of into shared eigenspaces ɡ of the adjoint operators as In most of books of Lie algebras we find the classification α theorem. In brief, there are the classical irreducible root ⊕ ɡα n ɡ = ɧ ∑ (4.2) systems A,,, BCDrepresented in as: α∈Φ n−1 nnn  where α∈Φ⊆ d are eigenvalues of adjH on the An−1 =−≠{ ee ij},, ij  j

eigenspace ɡα , in particular αi is the eigenvalue for Dnn= A−1  ±+( ee ij) ,, ij ≠ { } adjHi on ɡα . BD= ± e, nn{ i} ɡ  For each E∈=α ,, HHijα i E, α are called the CD= ± 2, e nn{ i} roots of ɡ.

200 M-Alamin A. H. Ahmed: The Role of Root System in Classification of Symmetric Spaces

Also we can write 5.1. Lie Groups and Lie Algebras in Symmetric Spaces

adjHi X= α ( H) X (4.3) A symmetric space can be represented as a coset space. i Also a symmetric space is associated to what is called an Where the root α∈ ɧ ∗ (the dual space of ɧ ). involutive automorphism of a given Lie algebra. Several different involutive automorphisms can act on the same 4.1.1. Theorem [3] algebra, so we normally have several different symmetric spaces deriving from the same Lie algebra. Every semisimple Lie algebra over contains a Cartan  If the field of the Lie algebra is the field of real, complex subalgebra. or quaternion numbers, the Lie algebra is called a real, 4.2. The Extended Basis for the Lie Algebra ɡ comlex or quaternion algebra. The classical Lie algebras sU( n+1,) , sO(2 n+ 1,) Let ɡα be the span of Eα for each α∈Φ (the set of sP(2, n) & sO( 2, n ) correspond to root systems root system). Then we may extend the basis {H1 ,...,Hr } ABC,,& D respectively. Also we have the five for ɧ into a basis {HE,...,H} { :α∈Φ} . nnn n 1 r α exceptional algebras corresponding to root systems For ɡ that satisfies the commutator relations GFEE2467,,, & E 8. HH,0= and HE, =α E, so we reach the ij [ iiαα] Each of these complex algebras in general has several real following fact, which can be shown by using the Killing forms associated to it. These real forms correspond to the form of ɡ, for more details see [1] & [3]. same Dynkin diagrams [3] and root systems as the complex algebras. 4.3. Fact [3] The semi (simple) complex algebra ɡ decomposes into a ɡ direct sum of root spaces: For the basis {HE1 ,...,Hr } { α :α∈Φ} of , the ɡ ɧ ɡ ɡ structure constants are: = ⊕∑ gα , where αα={E } , . −−αα={E } α  HHij,0= In general for any simple Lie algebra, the commutation relations determine the Cartan subalgebra and raising and [HEii, αα]=α E lowering operators, that in turn determine a unique root r system, and correspond to a given . In this = α EEαα, − ∑ ii H way we can classify all the simple algebras according to the = i 1 type of root system it possesses. 4,(αβ)  Eαβ+ , αβ+ ∈Φ 5.1.1. An Involutive Automorphism of a Lie Algebra = 22 EEαβ,  αβ ɡ  Let be a Lie algebra of the Lie group G . The mapping  0,αβ+ ∉Φ σ : ɡ → ɡ that preserves the algebraic operations on ɡ is called an automorphism of ɡ. If σ is linear automorphism satisfying σσ2 =Id( ≠ Id ) , so σ has eigen values 5. Symmetric Spaces ±1 , it splits the algebra ɡ into orthogonal eigenspaces These are spaces which possess the properties of corresponding to these eigen values. This mapping σ is symmetry and homogeneousness, they have many called an involutive automorphism. applications, this is because they have mixed algebraic and geometric properties. The beginning for these spaces is that 5.1.2. Complexification and of a Lie algebra they are spaces with parallel curvature tensor, later they were A complexification of a real Lie algebra is obtained by introduced through different approaches. They have much in taking linear combinations of its elements with complex common. Any symmetric space has its own special geometry, coefficients. The real Lie algebra ƞ is a real form of the Euclidean, elliptic and hyperbolic are some of these complex algebra ɡ if ɡ is the complexification of ƞ. geometries. They were first classified by Cartan who gave eleven classes of symmetric spaces in his classification. For 5.2. Symmetric Spaces as Coset Spaces more details of symmetric spaces see [2] & [3]. If ɡ is a , σ an involutive In this paper we disclose the relation between Lie algebras, ɡ root systems and symmetric spaces. Then we reach some automorphism of , and results. The restricted root systems are associated to ɡ = k ⊕ p where: symmetric spaces, just like ordinary root systems are associated to groups.

American Journal of Mathematics and Statistics 2016, 6(5): 197-202 201

5.2.2. Example σ ( X)= X, for X ∈k, G/ k = SU( n,/) SO( n ,) is a symmetric space of σ ( X)=−∈ X, for X (4.2.1) compact type and the related symmetric space of so k is a subalgebra, but p is not. The following commutation non-compact type is G ∗ / k = SL( n,/) SO( n ,) . relations hold: 5.3. Normal and Compact Real Forms of the Complex [k, k ] ⊂ k, [k, p] ⊂ p, [p, p] ⊂ k (4.2.2) Lie Algebra The subalgebra k satisfying equation (4 – 2) is called a symmetric subalgebra. The normal real form of a consists of the subspace in which the coefficients in its decomposition to 5.2.1. The Weyl Unitary Trick and Cartan Decomposition cartan subalgebra and other subspace are real. On the other hand, the compact real form of the complex Lie algebra is If we multiply elements in p mentioned above by i ( the obtained from the real form by the Weyl unitary trick. imaginary unit ), this is called weyl unitary trick, so we Classification of all the real forms of any complex Lie construct a new non- compact algebra. algebra can be done by enumeration of all involutive ɡ ∗ = k ⊕ ip. This is called a Cartan decomposition, and automorphisms of its compact form. k is a maximal compact subalgebra of ɡ ∗ . The coset spaces G/ k and G ∗ / k are symmetric spaces.

Table 1

Root space Restricted root space Cartan classes G / K G ∗ / K SL( n,) A − SU( n) n 1 A SU( n)

SU( n) SL( n,) An −1 AI SO( n) SO( n)

An −1 SU(2 n) SU∗ (2 n) An −1 AII USp(2 n) USp(2 n)

BCq ( p q) SU( p+ q) SU( p+ q) AIII C( pq= ) SU( p)×× SU( q) U (1) SU( p)×× SU( q) U (1) q SO(2 n+ 1, ) Bn Bn B SO(21 n+ ) SO(21 n+ )

Sp(2, n ) C C USp(2 n) n USp(2 n )

USp(2 n ) USp(2 n ) C Cn CI n SU( n)× U (1) SU( n)× U (1)

BCq ( p q) USp(22 n+ q ) USp(22 n+ q ) CII USp22 p× USp q USp22 p× USp q Cq ( pq= ) ( ) ( ) ( ) ( )

SO(2, n ) Dn D SO(2 n ) SO(2 n )

SO(4 n ) SO* (4 n ) Dn Cn DIII− even SU(21 n)× U ( ) SU(21 n)× U ( )

SO(42 n+ ) SO* (42 n+ ) BCn DIII− odd SU(21 n+×) U ( 1) SU(21 n+×) U ( 1)

Bn ( pq+=21 n +) Bpqq (  ) SO( p+ q) SO( p, q) BDI SO p× SO q SO p× SO q Dn ( pq+=2 n) Dq ( pq= ) ( ) ( ) ( ) ( )

202 M-Alamin A. H. Ahmed: The Role of Root System in Classification of Symmetric Spaces

5.3.1. Example the algebra sO(3 ) , the sphere SO(3/) SO( 2) and the The normal real form of the complex algebra ɡ c = double – sheeted hyperboloid SO(2,1) / SO( 2) . sL( n,) is the non – compact algebra ɡ ∗ = sL( n,) where ɡ ∗ = k ⊕ ip, k is the algebra of real, skew – symmetric and traceless nn× matrices. This algebra from 6. Conclusions the compact real form ɡ c = sL( n,) = k ⊕ p. - Classification of Lie algebras is an important tool in classification of symmetric spaces. 5.4. - Root systems give the basic classes of classification of symmetric spaces As a Lie algebra corresponds to a given root system, each - Compact and non-compact symmetric spaces can be symmetric space corresponds to a restricted root system, discussed by using the algebraic approach to these where these restricted root systems are important in some spaces, namely as a coset spaces of Lie groups and their physical applications. In some texsts these roots are often Lie algebras, then by using associated root systems we referred to in tables without explicitly mentioned that they can apply the classification machinery. are restricted. Generally the restricted root systems will be different from the original, inherited root system if the Cartan subalgebra lies in k (the symmetric subalgebra). To find the restricted root system we define an alternative Cartan subalgebra that REFERENCES lies partly (or entirely) in p (or ip) where p is a subspace of ɡ [1] R. Gilmore, Lie groups, Lie algebras, and some of their the algebra . For more details of restricted root systems see applications (John Wiley & Sons, New York 1974) ISBN: [1] & [3]. 0-471-30179-5. The following table discloses restricted root spaces associated to Cartan classes of symmetric spaces (see Table [2] W. M. Boothby, An Introductionto differentiable manifolds and Riemannian Geometry (Academic Press, New York 1). 1975). Finally we give the following explanatory example of symmetric spaces. [3] S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces (Academic Press, New York 1978) ISBN: 5.4.1. Example 0-471-30179-5. The algebra ɡ= sO(3, ) has a root system of type B , [4] J. Humphrey, “Introduction to Lie Algebras and  n ”. its compact real form is sO(3,) , and its only non – [5] J. Humphrey, “ Reflection groups and Coxeter groups”. compact real form is [6] Karin Erdmann and Mark J Wildson. Introduction to Lie sO( p,, q) sO( q ,, p ) , p+= q 3, obtained by algebras. Springer, 2006. σ = applying the involution IIpq,,,( qp ) to sO(3,) . [7] William Fulton and Joe Harris. Representation theory : a first There are two Riemannian symmetric spaces associated with course, volume 129. Springer, 1991.