American Journal of Computational and Applied Mathematics 2016, 6(5): 182-186 DOI: 10.5923/j.ajcam.20160605.02
On Extracting Properties of Lie Groups from Their Lie Algebras
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 The study of Lie groups and Lie algebras is very useful, for its wider applications in various scientific fields. In this paper, we introduce a thorough study of properties of Lie groups via their lie algebras, this is because by using linearization of a Lie group or other methods, we can obtain its Lie algebra, and using the exponential mapping again, we can convey properties and operations from the Lie algebra to its Lie group. These relations helpin extracting most of the properties of Lie groups by using their Lie algebras. We explain this extraction of properties, which helps in introducing more properties of Lie groups and leads to some important results and useful applications. Also we gave some properties due to Cartan’s first and second criteria. Keywords Lie group, Lie algebra, Exponential mapping, Linearization Killing form, Cartan’s criteria
1. Introduction 2. Lie Group The Properties and applications of Lie groups and their 2.1. Definition (Lie group) Lie algebras are too great to overview in a small paper. Lie groups were greatly studied by Marius Sophus Lie (1842 – A Lie group G is an abstract group and a smooth 1899), who intended to solve or at least to simplify ordinary n-dimensional manifold so that multiplication differential equations, and since then the research continues G×→ G G:,( a b) → ab and inverse G→→ Ga: a−1 to disclose their properties and applications in various are smooth. scientific fields [1, 2, 5, 6]. Lie groups have algebraic properties derived from their group axioms, and geometric 2.2. Examples properties due to identification of their group operations with points in a topological space, namely every Lie group is a a) The general linear group GL( n,)= differentiable manifold [3, 4, 7]. Almost all of the Lie groups ∈=≠nn× nn× encountered in many applications are matrix groups, and this {AA :det 0} is an open set of , it simplifies their study. This simplification can be done by is a manifold with the two operations, matrix linearizing the Lie group in the neighborhood of its identity, multiplication and matrix inverse which are C∞ which results in what is called a Lie algebra [1]. The Lie mappings. So it is a Lie group. algebra retains most of, but not all of the properties of the original Lie group. Also most of the Lie group properties can b) The complex linear group GL( n,) , the group of be recovered by the inverse of the linearization operation, nonsingular nn× matrices is also a Lie group. namely, the exponential mapping. A Lie algebra is a linear c) We can take the two subsets of the general linear vector space, and this makes it easy to study it by using tools of linear algebra rather than studying the original Lie group group: the special linear group SL( n,) which is with its complicated structure. In this manner we can extract the group of matrices of determinant 1 and the other is properties of a Lie group from its Lie algebra, and this opens the orthogonal group On( ) consisting of all the door to many scientific applications, for instance in T mathematics and physics. matrices A satisfying AA= I, these two subsets are also Lie groups. * Corresponding author: [email protected] (M-Alamin A. H. Ahmed) Matrix groups are very important examples of Lie groups Published online at http://journal.sapub.org/ajcam because of their various scientific applications. There are Copyright © 2016 Scientific & Academic Publishing. All Rights Reserved many examples of Lie groups different from matrix groups.
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3. Lie Algebra linearization immediately as we see in the following example: 3.1. Definition (Lie Algebra) 4.2. Example A Lie algebra over a field K ( or ) is a vector space V over K with a skew – symmetric k- bilinear form (the Lie To linearize the Lie group bracket) [ ,:] VV×→ V, which satisfies the Jacobi identity: SL(2,)= {M∈= GL(2,) ,det M 1}
X,,[ YZ] ++= Y ,,[ ZX] Z ,[ XY ,] 0 (3.1) We parametrize the group by: 1+ab For all XYZ,, ∈ V →= (4.1) (abc,,) M( abc ,,) (1+bc) The tangent space TGe at the identity of a Lie group c (1+ a) canonically has the structure of Lie algebra. This Lie algebra encodes in it much information about the Lie group. So we then we let (abc,,) to become infinitesimals: have to define the Lie algebra structure on TGe and this can abc,,→→δδδ abc ,, M δδδ abc ,, be shown if we identified this structure with the Lie algebra ( ) ( ) ( ) for instance for some classical groups. 1+δδab (4.2) = (1+δδbc) 3.2. Example δ c (1+δ a) × Since GL( n,) is an open subset of nn (the vector so the basis vectors in the Lie algebra X, X and X are space of all nn× real matrices), the tangent space to ab c × the coefficients of the first order infinitesimals: GL( n,) at the identity I is nn itself. 1+δδab Also the tangent space TI ( SL( n,)) is the subspace of (δδδa,, b c)→+ I2 δ aXabc + δ bX + δ cX =(4.3) δδca1− nn× × consisting of all nn matrices of trace 0. We get the basis vectors from (4-1) as follows: And for the group of orthogonal matrices On( ) , the tangent space at the identity is consisting of skew – ∂M 10 symmetric matrices. These are some examples of Lie X a = = , ∂a (0,0,0) 01− algebras of some groups of matrices considered as tangent spaces at the identity. ∂M 01 (4.4) Xb = = , ∂b (0,0,0) 00 4. Linearization of a Lie Group ∂M 00 X = = The Lie group may have a complicated structure which c ∂c (0,0,0) 10 make it difficult to study it through some mathematical tools, instead of this we linearize the group in the neighborhood of These are the basis vectors for the Lie algebra sl (2,) its identity, that is by getting a vector space at the identity, this vector space is its Lie algebra. Therefore, Linearization in our example. of Lie group makes it possible to study it by using tools of Linear algebra. Moreover most of the Lie group properties can be recovered by the inverse of the linearization operation, 5. Exponential Mapping that is by the exponential mapping. The following diagram Suppose V is a finite dimensional vector space equipped expresses how to go from the Lie group to its Lie algebra and with a complete norm . over the field or . If vice versa by using these two operations: End( V ) denotes the algebra of linear automorphisms on , Linearization V Lie group ←→ Lie alg ebra n Exponentiation and GL( V ) the general linear group. Then for V = ,
End( V ) is identified with M n () ( nn× matrices ) and 4.1. How To Linearize a Lie Group? GL( V)= GL( n,) . In order to linearize a Lie group to get its Lie algebra we need an explicit parametrization of the underlying manifold 5.1. Definition (Matrix Group) and also we need an available expression for the group composition law. For matrix groups the group composition A matrix group is a closed subgroup of GL( V ) (the law is matrix multiplication and every matrix group is a general linear group), where V is a finite dimensional closed subset of the general linear group, so we can do vector space.
184 M-Alamin A. H. Ahmed: On Extracting Properties of Lie Groups from Their Lie Algebras
5.2. Definition (Exponential map) 5.10. Theorem (The Lie Algebra of a Matrix Group) The exponential map on End( V ) is defined by For a matrix group G⊆ GL( V ) , the set
∞ n =∈∈∈ A g{ A EndV:exp( tA) G for all t R} is a Lie algebra, exp A =, A ∈ End V (5.1) ( ) ∑ ( ) called the Lie algebra of G . n =0 n! Sketch of the Proof: If G is a Lie group with Lie algebra g , then the The theorem can be proved by showing that: exponential map is defined as: exp(t[ A , B])= exp([ tA , B]) ∈∈ G for all t . Thus the exp:g→= GwhereexpXφφ 1 with' 0 = X , ( ) XX( ) ( ) commutator [AB, ]∈ g. ∀∈Xg 5.11. Computing Lie Algebras 5.3. Definition (One – Parameter Subgroup) For matrix groups we can use some methods to find their A one – parameter subgroup of a Lie group G is a Lie algebras. For instance we can use the following continuous homomorphism: α :,( +→) G proposition for this purpose.
5.4. Example 5.12. Proposition For the Lie groups GL( n,,) and GL( n ) , we have Let β (.,.) be a continuous bilinear form on the vector (ts, ) exp( A)= eA , e = etA e sA ,.. ieφ ( t)= etA is a one parameter space V and set G=∈{ h GL( V) :, ∀∈ x y V , ' A ββ= group. Furthermore φφ(0)= A and henceexp( A)=( I) = e . (hx,hy) ( x , y)} , then the Lie algebra is g=∈{ A EndV:, ∀∈ x y V ,ββ( Ax , y) +( x , Ay) = 0} 5.5. Proposition [2] Proof For V a finite dimensional normed vector space and ∈ β = A∈ End( V ) , the map t→exp( tA) is a one – parameter Let Ag, then (exp(tA) x ,exp( tA) y) −1 β x,, y for all x y∈ V . subgroup of GL( V ) . In particular (exp( AA)) = exp( − ) . ( ) Differentiating with respect to t: 5.6. Theorem [2] ββ( Aexp( tA) x ,exp( tA) y)+=( exp( tA) x ,exp A( tA) y) 0 , Every one – parameter subgroup α :→End( V ) is of then at t =0 ⇒ ββ( Ax, y)+=( x ,0 Ay) . the form α (t)=exp( tA) for some A∈ End( V ). Conversely: for all x,, y∈ Vββ( Ax , y) +=( x , Ay) 0.
5.7. Definition (The Infinitesimal Generator) Then for f( t)=β (exp( tA) x ,exp( tA) y) , f′( t)= 0 , f The element A∈ End( V ) is called the infinitesimal therefore is constant function of a value β x,0 y at t = generator of the one – parameter subgroup t→exp( tA) . ( ) . It follows that: exp(tA)∈∈ G for all t, that is A g 5.8. Lemma The exponential map is absolutely convergent, hence convergent on all of End( V ) . Hence it defines an analytic 6. Exponentiation automorphism on End( V ) We can reverse the linearization operation by using the exponential map so as to recover most of the properties of the Proof Lie group from its Lie algebra. We can see the behavior of n ∞∞An A elements near the identity of the Lie group. Suppose X is ∑∑≤=exp( A ) an operator in a Lie algebra, and let δ be a small real nn=00nn!!= number. Then I +δ represents an element in the Lie group close to the identity. Far from the identity by iterating this 5.9. Lemma [2] group operation many times we get the exponential function: k The identity map on End( V ) is the derivative at 0 of 1 ∞ X n lim IX+=∑ =exp( X) (6.1) exp :EndVV→ End = k→∞ ( ) ( ) that is, dexp( 0) Id . knn =0 !
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6.1. Example The Lie algebras are isomorphic because there is a 1:1 correspondence between them, but the groups have a 2:1 On taking the Lie group G= SL(2, ) , we can show how correspondence, that is why they are not isomorphic. to exponentiate an operator in its Lie algebra to get the result in the group G as follows:
given the basis vectors Xabc, X , X∈ sl ( 2,) (the Lie 8. Some Properties algebra), let us take the operator 8.1. Definition (Ideals and Simple Lie Algebras) ab X=++= aXabc bX cX , then An ideal I of a Lie algebra ɡ is a vector space of ɡ such ca− that [a,, b]∈ I ∀∈ a I and b ∈ɡ n ∞ 1 ab A simple Lie algebra is the one which has no proper ideal. = ++ = exp( X) exp( aXabc bX cX ) ∑ Also a semisimple Lie algebra is the one which is a direct = n!ca− n 0 sum of simple Lie algebras. sinhθ =IXcoshθ + 2 θ 8.2. Example abθθ (6.2) Let sl be the set of all nn× matrices of trace 0. θ + sinh( ) sinh( ) n () cosh θθ =sl () is an ideal of gl () which is nonzero. So casinh(θθ) sinh( ) n n coshθ − θθgln () is not simple. where 22 . θ =a + bc 8.3. Definition (The Adjoint Representation) 0 1 22 In fact X= I22,, X = XX =θ I so Let ɡ be a Lie algebra over k. A representation of ɡ is a 3 21 2 4 31 X= XX =θ X is proportional to X and X= XX Lie algebra homomorphism ρ:,ɡ→ gl( n k ) for some n is proportional to the identity and so on. called the degree of the representation. We define a 01 We can put F( X)= f01( abc,,) X+ f( abc ,,) X mapping ad X from a Lie algebra to itself by where ff01& are functions of the invariants of the matrix, adX:, Y→[ X Y ] . The mapping X→ ad X is a that is θ 22=a + bc (Cayley – Hamilton theorem). representation of the Lie algebra called the adjoint representation. It is an automorphism. So that 22 8.4. Cartan Subalgebra ɧ exp( Xf)=0(θθ) If 21 + ( ) X (6.3) A cartan subalgebra ɧ for a Lie algebra ɡ is an abelian, where diagonalizable subalgebra which is maximal under set ɡ θθθ246 inclusion. Its dimension is the rank of . 2 ɡ fI0 (θθ)=++++=... cosh (6.4) All Cartan Lie algebras of a Lie algebra are conjugate 2! 4! 6! under automorphisms of ɡ , and they have the same θθθ246 sinh(θ ) dimension. fI1 (θ )=++++=... (6.5) Define the basis {H ,...,H } for ɧ. Since ɧ is abelian, 3! 5! 7! θ 1 r HHij,0= for all ij, . 7. Isomorphism on Lie Groups and Lie If we extend this basis for ɧ to a basis for ɡ, and then we get a much simpler basis for ɡ with convenient Algebras commutator relations. This can be used in many Isomorphic Lie groups have isomorphic Lie algebras, but, applications of Lie algebras, especially in classification two Lie groups with isomorphic Lie algebras need not be machinery. isomorphic as we see in the following example. 8.5. The Killing Form 7.1. Example For the Lie algebra g, if XY, ∈ g, we define the Killing The groups SO(2,1) & SU ( 1,1) have respectively the form by k( X, Y)= tr adj X adjY , it resembles a scalar Lie algebras product for elements of a Lie algebra itself. It is invariant under all automorphisms of g. 0 aa32 i b ib+ b The killing form is an effective tool in calculations in Lie =−=3 12. so(2,1) a31 0 a & su( 1,1) algebra and symmetric spaces and in describing some 2 ib12−− b b 3 aa210 properties due to some Cartan’s theorems.
186 M-Alamin A. H. Ahmed: On Extracting Properties of Lie Groups from Their Lie Algebras
8.6. Cartan’s First Criterion [8] most of these properties can be retained. A Lie algebra g is solvable if and only if its Killing form 4. Since isomorphic Lie groups have isomorphic Lie algebras, research can go on studying joined vanishes identically on the derived Lie algebra g′. properties of Lie groups using their isomorphic Lie It is easy to see that both solvability and vanishing of the algebras. Killing form remain unchanged under complexification for a 5. The killing form is an effective tool in describing real Lie algebra g; thus we may take g complex. some important properties of the Lie algebra and in 8.7. Cartan’s Second Criterion [8] carrying some calculations on it. A Lie algebra g is semisimple if and only if its dimension is positive and its killing form is non-degenerate. (Non-degenerate means: If for some X0 in g the value of the killing form is 0 for all Y in g, then X0 is 0). REFERENCES Just as for the first criterion we may assume that g is [1] Robert Gilmore, Lie Groups, Physics, and Geometry, complex, since both semisimplicity and non-degeneracy of Cambridge University Press, 2008. the killing form are unchanged by complixification. [2] Loring W. Tu,, An Introduction to Manifolds, Springer, 2008. [3] William M. Boothby, An Introduction to Differentiable 9. Conclusions Manifolds and Riemannian Geometry, Academic Press Inc. New York, 1975. 1. The exponential operation provides a natural parametrization of the Lie group in terms of linear [4] Jimmie Lawson, Matrix Lie Groups and Control Theory, Summer 2007. quantities. This function maps the linear vector space – the Lie algebra – to the geometric manifold that [5] Sigurdur Helgason, Differential Geometry, Lie Groups and parameterizes the Lie group. Symmetric Spaces, Academic Press INC. 1978. 2. It is not always possible to map a Lie algebra onto the [6] William Fulton and Joe Harris. Representation theory: a first entire Lie group through a single mapping of the form course, volume 129. Springer, 1991. exp(X). 3. Since the exponential map takes elements of the Lie [7] Karin Erdmann and Mark J Wildson. Introduction to Lie algebras. Springer, 2006. algebra to the geometric manifold that parametrizes the Lie group, we expect not to be able to extract all [8] Hans Samelson, Notes on Lie Algebras, third edition. properties of the Lie group from its Lie algebra, but