On Extracting Properties of Lie Groups from Their Lie Algebras

Transcription

1 America Joural of Computatioal ad Applied Mathematics 216, 6(5): DOI: /j.ajcam O Extractig Properties of Lie Groups from Their Lie Algebras M-Alami A. H. Ahmed 1,2 1 Departmet of Mathematics, Faculty of Sciece ad Arts Khulais, Uiversity of Jeddah, Saudi Arabia 2 Departmet of Mathematics, Faculty of Educatio, Alzaeim Alazhari Uiversity (AAU), Khartoum, Suda Abstract The study of Lie groups ad Lie algebras is very useful, for its wider applicatios i various scietific fields. I this paper, we itroduce a thorough study of properties of Lie groups via their lie algebras, this is because by usig liearizatio of a Lie group or other methods, we ca obtai its Lie algebra, ad usig the expoetial mappig agai, we ca covey properties ad operatios from the Lie algebra to its Lie group. These relatios helpi extractig most of the properties of Lie groups by usig their Lie algebras. We explai this extractio of properties, which helps i itroducig more properties of Lie groups ad leads to some importat results ad useful applicatios. Also we gave some properties due to Carta s first ad secod criteria. Keywords Lie group, Lie algebra, Expoetial mappig, Liearizatio Killig form, Carta s criteria 1. Itroductio The Properties ad applicatios of Lie groups ad their Lie algebras are too great to overview i a small paper. Lie groups were greatly studied by Marius Sophus Lie ( ), who iteded to solve or at least to simplify ordiary differetial equatios, ad sice the the research cotiues to disclose their properties ad applicatios i various scietific fields [1, 2, 5, 6]. Lie groups have algebraic properties derived from their group axioms, ad geometric properties due to idetificatio of their group operatios with poits i a topological space, amely every Lie group is a differetiable maifold [3, 4, 7]. Almost all of the Lie groups ecoutered i may applicatios are matrix groups, ad this simplifies their study. This simplificatio ca be doe by liearizig the Lie group i the eighborhood of its idetity, which results i what is called a Lie algebra [1]. The Lie algebra retais most of, but ot all of the properties of the origial Lie group. Also most of the Lie group properties ca be recovered by the iverse of the liearizatio operatio, amely, the expoetial mappig. A Lie algebra is a liear vector space, ad this makes it easy to study it by usig tools of liear algebra rather tha studyig the origial Lie group with its complicated structure. I this maer we ca extract properties of a Lie group from its Lie algebra, ad this opes the door to may scietific applicatios, for istace i mathematics ad physics. * Correspodig author: modhsd@gmail.com (M-Alami A. H. Ahmed) Published olie at Copyright 216 Scietific & Academic Publishig. All Rights Reserved 2. Lie Group 2.1. Defiitio (Lie group) A Lie group G is a abstract group ad a smooth -dimesioal maifold so that multiplicatio G G G :( a, b) ab ad iverse 1 G G: a a are smooth Examples a) The geeral liear group GL( ) { :det = }, = A A is a ope set of, it is a maifold with the two operatios, matrix multiplicatio ad matrix iverse which are C mappigs. So it is a Lie group. b) The complex liear group GL(, ), the group of osigular matrices is also a Lie group. c) We ca take the two subsets of the geeral liear SL, which is group: the special liear group the group of matrices of determiat 1 ad the other is the orthogoal group O( ) cosistig of all matrices A satisfyig T A A= I, these two subsets are also Lie groups. Matrix groups are very importat examples of Lie groups because of their various scietific applicatios. There are may examples of Lie groups differet from matrix groups.

2 America Joural of Computatioal ad Applied Mathematics 216, 6(5): Lie Algebra 3.1. Defiitio (Lie Algebra) A Lie algebra over a field K ( or ) is a vector space V over K with a skew symmetric k- biliear form (the Lie, : V V V, which satisfies the Jacobi idetity: bracket) [ ] [ ] [ ] [ ], Y, Z + Y, Z, + Z,, Y = (3.1) For all, Y, Z V The taget space TG e at the idetity of a Lie group caoically has the structure of Lie algebra. This Lie algebra ecodes i it much iformatio about the Lie group. So we have to defie the Lie algebra structure o TG e ad this ca be show if we idetified this structure with the Lie algebra for istace for some classical groups Example GL is a ope subset of (the vector space of all real matrices), the taget space to GL(, ) at the idetity I is itself. Also the taget space TI ( SL(, )) is the subspace of cosistig of all matrices of trace. O, the Sice (, ) Ad for the group of orthogoal matrices taget space at the idetity is cosistig of skew symmetric matrices. These are some examples of Lie algebras of some groups of matrices cosidered as taget spaces at the idetity. 4. Liearizatio of a Lie Group The Lie group may have a complicated structure which make it difficult to study it through some mathematical tools, istead of this we liearize the group i the eighborhood of its idetity, that is by gettig a vector space at the idetity, this vector space is its Lie algebra. Therefore, Liearizatio of Lie group makes it possible to study it by usig tools of Liear algebra. Moreover most of the Lie group properties ca be recovered by the iverse of the liearizatio operatio, that is by the expoetial mappig. The followig diagram expresses how to go from the Lie group to its Lie algebra ad vice versa by usig these two operatios: Liearizatio Lie group Lie algebra Expoetiatio 4.1. How To Liearize a Lie Group? I order to liearize a Lie group to get its Lie algebra we eed a explicit parametrizatio of the uderlyig maifold ad also we eed a available expressio for the group compositio law. For matrix groups the group compositio law is matrix multiplicatio ad every matrix group is a closed subset of the geeral liear group, so we ca do liearizatio immediately as we see i the followig example: 4.2. Example To liearize the Lie group 2, = M GL 2,,det M = 1 SL { } We parametrize the group by: a b ( abc,, ) M( abc,, ) = ( 1+ bc) c ( 1+ a ) the we let ( abc,, ) to become ifiitesimals: ( a, b, c) ( δa, δb, δc) M( δa, δb, δc) (4.1) δa δb (4.2) = ( 1+ δbδc) δ c ( 1+ δ a ) so the basis vectors i the Lie algebra a, bad c are the coefficiets of the first order ifiitesimals: δa δb δa δb δc I + δa a+ δbb+ δc c= δ 1 δ c a We get the basis vectors from (4-1) as follows: (,, ) 2 a b c 1 = =, a (,,) 1 1 = =, b (,,) = = c (,,) 1 (4.3) (4.4) These are the basis vectors for the Lie algebra sl ( 2, ) i our example. 5. Expoetial Mappig Suppose V is a fiite dimesioal vector space equipped with a complete orm. over the field or. If Ed ( V ) deotes the algebra of liear automorphisms o V, ad GL( V ) the geeral liear group. The for V =, Ed ( V ) is idetified with M ( ) ( matrices ) ad GL( V ) = GL(, ) Defiitio (Matrix Group) A matrix group is a closed subgroup of GL V (the geeral liear group), where V is a fiite dimesioal vector space.

3 184 M-Alami A. H. Ahmed: O Extractig Properties of Lie Groups from Their Lie Algebras 5.2. Defiitio (Expoetial map) The expoetial map o Ed ( V ) is defied by A =! ( A) = A Ed ( V ) exp, (5.1) If G is a Lie group with Lie algebra g, the the expoetial map is defied as: ' φ φ exp :g G where exp = 1 with =, g 5.3. Defiitio (Oe Parameter Subgroup) A oe parameter subgroup of a Lie group G is a α :, + G cotiuous homomorphism: 5.4. Example For the Lie groups GL(, ) ad GL(, ), we have A ( t, s) ta sa ta exp ( A) = e, e = e e, ie.. φ ( t) = e is a oe parameter ' group. Furthermore φ ( ) = A ad hece exp( A) = φ( I ) = e A Propositio [2] For V a fiite dimesioal ormed vector space ad t exp ta is a oe parameter A Ed ( V ), the map 1 subgroup of GL( V ). I particular ( ( A )) = ( A ) 5.6. Theorem [2] exp exp. Every oe parameter subgroup α : the form α =exp t ta for some A Ed ( V ) Defiitio (The Ifiitesimal Geerator) Ed V is of The elemet A Ed ( V ) is called the ifiitesimal geerator of the oe parameter subgroup t exp ( ta ) Lemma The expoetial map is absolutely coverget, hece Ed V. Hece it defies a aalytic coverget o all of automorphism o Ed ( V ) Proof 5.9. Lemma [2] A A = exp!! = = ( A ) The idetity map o Ed ( V ) is the derivative at of ( V) Ed( V ) that is, d exp( ) = Id. exp :Ed 5.1. Theorem (The Lie Algebra of a Matrix Group) For a matrix group G GL( V ), the set = :exp { } g A EdV ta G for all t R is a Lie algebra, called the Lie algebra of G. Sketch of the Proof: The theorem ca be proved by showig that: exp t A, B = exp ta, B G for all t. Thus the ( [ ]) ([ ]) commutator [ AB, ] g Computig Lie Algebras For matrix groups we ca use some methods to fid their Lie algebras. For istace we ca use the followig propositio for this purpose Propositio Let β (.,.) be a cotiuous biliear form o the vector space V ad set G = { h GL( V ) : x, y V, β( hx,hy ) = β( x, y )}, the the Lie algebra is g = { A EdV : x, y V, β( Ax, y) + β( x, Ay ) = } Proof Let A g, the β exp ( ta) x,exp( ta) y = β x, y for all x, y V. Differetiatig with respect to t: β Aexp ta x,exp ta y + β exp ta x, Aexp ta y =, ( ) ( ) t β( Ax, y) + β( x, Ay ) =. Coversely: for all β β The for β ( ) the at = x, y V, Ax, y + x, Ay =. f t = exp ta x,exp ta y, f t =, therefore f is costat fuctio of a value β x, y at t =. It follows that: ( ta) exp G for all t, that is A g 6. Expoetiatio We ca reverse the liearizatio operatio by usig the expoetial map so as to recover most of the properties of the Lie group from its Lie algebra. We ca see the behavior of elemets ear the idetity of the Lie group. Suppose is a operator i a Lie algebra, ad let δ be a small real umber. The I + δ represets a elemet i the Lie group close to the idetity. Far from the idetity by iteratig this group operatio may times we get the expoetial fuctio: k 1 lim I+ = = exp k k! = (6.1)

4 America Joural of Computatioal ad Applied Mathematics 216, 6(5): Example O takig the Lie group = ( 2, ) G SL, we ca show how to expoetiate a operator i its Lie algebra to get the result i the group G as follows: give the basis vectors a, b, c sl ( 2, ) (the Lie algebra), let us take the operator a b = a a+ bb+ c c= c a, the exp 1 a b = a+ b+ c =! c a ( ) exp( a b c ) = sih = I2 cosh + asih( ) bsih( ) cosh + = sih( ) sih( ) c a cosh (6.2) where = a + bc. 1 I fact = I2, =, = I 2 so = = is proportioal to ad = is proportioal to the idetity ad so o. We ca put F( ) = f 1 abc,, + f1 abc,, where f & f 1 are fuctios of the ivariats of the matrix, that is = a + bc (Cayley Hamilto theorem). So that where = 2+ 1 exp f I f (6.3) 2 ( ) f = I = cosh (6.4) 2! 4! 6! sih f1 ( ) = I = (6.5) 3! 5! 7! 7. Isomorphism o Lie Groups ad Lie Algebras Isomorphic Lie groups have isomorphic Lie algebras, but, two Lie groups with isomorphic Lie algebras eed ot be isomorphic as we see i the followig example Example SO SU have respectively the Lie algebras a a so a a su. The groups ( 2,1 )& ( 1,1) 3 2 i b3 ib1+ b2 = 3 1 = ib b b a a ( 2,1) & ( 1,1) The Lie algebras are isomorphic because there is a 1:1 correspodece betwee them, but the groups have a 2:1 correspodece, that is why they are ot isomorphic. 8. Some Properties 8.1. Defiitio (Ideals ad Simple Lie Algebras) A ideal I of a Lie algebra ɡ is a vector space of ɡ such that [ a, b] I, a I ad b ɡ A simple Lie algebra is the oe which has o proper ideal. Also a semisimple Lie algebra is the oe which is a direct sum of simple Lie algebras Example sl be the set of all matrices of trace. Let ( ) sl ( ) is a ideal of ( ) gl ( ) is ot simple. gl which is ozero. So 8.3. Defiitio (The Adjoit Represetatio) Let ɡ be a Lie algebra over k. A represetatio of ɡ is a Lie algebra homomorphism ρ : ɡ gl (, k ) for some called the degree of the represetatio. We defie a mappig ad from a Lie algebra to itself by ad : Y [, Y ]. The mappig ad is a represetatio of the Lie algebra called the adjoit represetatio. It is a automorphism Carta Subalgebra ɧ A carta subalgebra ɧ for a Lie algebra ɡ is a abelia, diagoalizable subalgebra which is maximal uder set iclusio. Its dimesio is the rak of ɡ. All Carta Lie algebras of a Lie algebra ɡ are cojugate uder automorphisms of ɡ, ad they have the same dimesio. H for ɧ. Sice ɧ is abelia, Hi, H j= for all i, j. If we exted this basis for ɧ to a basis for ɡ, ad the we get a much simpler basis for ɡ with coveiet commutator relatios. This ca be used i may applicatios of Lie algebras, especially i classificatio machiery. Defie the basis { } 1,...,H r 8.5. The Killig Form For the Lie algebra g, if, Y g, we defie the Killig form by k (, Y ) = tr adj adjy, it resembles a scalar product for elemets of a Lie algebra itself. It is ivariat uder all automorphisms of g. The killig form is a effective tool i calculatios i Lie algebra ad symmetric spaces ad i describig some properties due to some Carta s theorems.

5 186 M-Alami A. H. Ahmed: O Extractig Properties of Lie Groups from Their Lie Algebras 8.6. Carta s First Criterio [8] A Lie algebra g is solvable if ad oly if its Killig form vaishes idetically o the derived Lie algebra g. It is easy to see that both solvability ad vaishig of the Killig form remai uchaged uder complexificatio for a real Lie algebra g; thus we may take g complex Carta s Secod Criterio [8] A Lie algebra g is semisimple if ad oly if its dimesio is positive ad its killig form is o-degeerate. (No-degeerate meas: If for some i g the value of the killig form is for all Y i g, the is ). Just as for the first criterio we may assume that g is complex, sice both semisimplicity ad o-degeeracy of the killig form are uchaged by complixificatio. 9. Coclusios 1. The expoetial operatio provides a atural parametrizatio of the Lie group i terms of liear quatities. This fuctio maps the liear vector space the Lie algebra to the geometric maifold that parameterizes the Lie group. 2. It is ot always possible to map a Lie algebra oto the etire Lie group through a sigle mappig of the form exp(). 3. Sice the expoetial map takes elemets of the Lie algebra to the geometric maifold that parametrizes the Lie group, we expect ot to be able to extract all properties of the Lie group from its Lie algebra, but most of these properties ca be retaied. 4. Sice isomorphic Lie groups have isomorphic Lie algebras, research ca go o studyig joied properties of Lie groups usig their isomorphic Lie algebras. 5. The killig form is a effective tool i describig some importat properties of the Lie algebra ad i carryig some calculatios o it. REFERENCES [1] Robert Gilmore, Lie Groups, Physics, ad Geometry, Cambridge Uiversity Press, 28. [2] Lorig W. Tu,, A Itroductio to Maifolds, Spriger, 28. [3] William M. Boothby, A Itroductio to Differetiable Maifolds ad Riemaia Geometry, Academic Press Ic. New York, [4] Jimmie Lawso, Matrix Lie Groups ad Cotrol Theory, Summer 27. [5] Sigurdur Helgaso, Differetial Geometry, Lie Groups ad Symmetric Spaces, Academic Press INC [6] William Fulto ad Joe Harris. Represetatio theory: a first course, volume 129. Spriger, [7] Kari Erdma ad Mark J Wildso. Itroductio to Lie algebras. Spriger, 26. [8] Has Samelso, Notes o Lie Algebras, third editio.