Chapter 16 Lie Groups, Lie Algebras and the Exponential Map 16.1 Lie Groups and Lie Algebras Now that we have the general concept of a manifold, we can define Lie groups in more generality. If every Lie group was a linear group (a group of ma- trices), then there would be no need for a more general definition. However, there are Lie groups that are not matrix groups, although it is not a trivial task to exhibit such groups and to prove that they are not matrix groups. 765 766 CHAPTER 16. LIE GROUPS, LIE ALGEBRA, EXPONENTIAL MAP An example of a Lie group which is not a matrix group is the quitient group G = H/N,whereH (the Heisenberg group) is the group of 3 3uppertriangularmatrices given by ⇥ 1 ab H = 01c a, b, c R , 80 1 | 2 9 < 001 = @ A and N is the discrete: group ; 10k2⇡ N = 01 0 k Z . 80 1 | 2 9 < 00 1 = @ A Both groups H and:N are matrix groups, ;N is closed and norma, yet G = H/N is a Lie group and it can be shown using some representation theory that G is not a matrix group 16.1. LIE GROUPS AND LIE ALGEBRAS 767 Another example of a Lie group that is not a matrix group is obtained by considering the universal cover SL^(n, R) of SL(n, R)forn 2. ≥ The group SL(n, R)isamatrixgroupwhichisnotsimply connected for n 2, and its universal cover SL^(n, R)is aLiegroupwhichisnotamatrixgroup.≥ Definition 16.1. A Lie group is a nonempty subset, G,satisfyingthefollowingconditions: (a) G is a group (with identity element denoted e or 1). (b) G is a smooth manifold. (c) G is a topological group. In particular, the group operation, : G G G,andtheinversemap, 1 · ⇥ ! − : G G,aresmooth. ! Remark: The smoothness of inversion follows automat- ically from the smoothness of multiplication. This can be shown by applying the inverse function theorem to the map (g, h) (g, gh), from G G to G G. 7! ⇥ ⇥ 768 CHAPTER 16. LIE GROUPS, LIE ALGEBRA, EXPONENTIAL MAP We have already met a number of Lie groups: GL(n, R), GL(n, C), SL(n, R), SL(n, C), O(n), SO(n), U(n), SU(n), E(n, R). Also, every linear Lie group (i.e., a closed subgroup of GL(n, R)) is a Lie group. We saw in the case of linear Lie groups that the tangent space to G at the identity, g = T1G,playsaveryimpor- tant role. This is again true in this more general setting. Definition 16.2. A (real) Lie algebra, ,isareal vector space together with a bilinear map, [ , A]: ,calledtheLie bracket on such that· the· A following⇥A! twoA identities hold for all a, b, cA : 2A [a, a]=0, and the so-called Jacobi identity [a, [b, c]] + [c, [a, b]] + [b, [c, a]] = 0. It is immediately verified that [b, a]= [a, b]. − 16.1. LIE GROUPS AND LIE ALGEBRAS 769 For every a ,itiscustomarytodefinethelinearmap ad(a): 2Aby A!A ad(a)(b)=[a, b],b . 2A The map ad(a)isalsodenotedada or ad a. Let us also recall the definition of homomorphisms of Lie groups and Lie algebras. Definition 16.3. Given two Lie groups G1 and G2,a homomorphism (or map) of Lie groups is a function, f : G G ,thatisahomomorphismofgroupsanda 1 ! 2 smooth map (between the manifolds G1 and G2). Given two Lie algebras 1 and 2,ahomomorphism (or map) of Lie algebras Ais a function,A f : ,thatisa A1 !A2 linear map between the vector spaces 1 and 2 and that preserves Lie brackets, i.e., A A f([A, B]) = [f(A),f(B)] for all A, B . 2A1 770 CHAPTER 16. LIE GROUPS, LIE ALGEBRA, EXPONENTIAL MAP An isomorphism of Lie groups is a bijective function f 1 such that both f and f − are maps of Lie groups, and an isomorphism of Lie algebras is a bijective function 1 f such that both f and f − are maps of Lie algebras. The Lie bracket operation on g can be defined in terms of the so-called adjoint representation. Given a Lie group G,foreverya G we define left translation as the map, L : G G2,suchthat a ! La(b)=ab,forallb G,andright translation as the map, R : G G,suchthat2 R (b)=ba,forallb G. a ! a 2 Because multiplication and the inverse maps are smooth, the maps La and Ra are di↵eomorphisms, and their deriva- tives play an important role. The inner automorphisms R 1 La (also written R 1La a− ◦ a− or Ada)alsoplayanimportantrole.Notethat 1 Ad (b)=R 1L (b)=aba− . a a− a 16.1. LIE GROUPS AND LIE ALGEBRAS 771 The derivative d(Ad ) : T G T G a 1 1 ! 1 of Ad : G G at 1 is an isomorphism of Lie algebras, a ! denoted by Ad : g g. a ! The map Ad: G GL(g)givenbya Ad is a group ! 7! a homomorphism from G to GL(g). Furthermore, this map is smooth. Proposition 16.1. The map Ad: G GL(g) is smooth. Thus it is a Lie algebra homomorphism.! Definition 16.4. The map a Ada is a map of Lie groups 7! Ad: G GL(g), ! called the adjoint representation of G (where GL(g) denotes the Lie group of all bijective linear maps on g). 772 CHAPTER 16. LIE GROUPS, LIE ALGEBRA, EXPONENTIAL MAP In the case of a Lie linear group, we have verified earlier that 1 Ad(a)(X)=Ada(X)=aXa− for all a G and all X g. 2 2 Since Ad: G GL(g)issmooth,itsderivative ! dAd : g gl(g)exists. 1 ! Definition 16.5. The derivative dAd : g gl(g) 1 ! of Ad: G GL(g)at1ismapofLiealgebras,denoted by ! ad: g gl(g), ! called the adjoint representation of g. In the case of a linear group, we showed that ad(A)(B)=[A, B] for all A, B g. 2 16.1. LIE GROUPS AND LIE ALGEBRAS 773 One can also check (in general) that the Jacobi identity on g is equivalent to the fact that ad preserves Lie brackets, i.e., ad is a map of Lie algebras: ad([u, v]) = [ad(u), ad(v)], for all u, v g (where on the right, the Lie bracket is the 2 commutator of linear maps on g). In the case of an abstract Lie group G,sinceadisdefined, we would like to define the Lie bracket of g in terms of ad. This is the key to the definition of the Lie bracket in the case of a general Lie group (not just a linear Lie group). 774 CHAPTER 16. LIE GROUPS, LIE ALGEBRA, EXPONENTIAL MAP Definition 16.6. Given a Lie group, G,thetangent space, g = T1G,attheidentitywiththeLiebracket defined by [u, v]=ad(u)(v), for all u, v g, 2 is the Lie algebra of the Lie group G. Actually, we have to justify why g really is a Lie algebra. For this, we have Proposition 16.2. Given a Lie group, G, the Lie bracket, [u, v]=ad(u)(v), of Definition 16.6 satisfies the axioms of a Lie algebra (given in Definition 16.2). Therefore, g with this bracket is a Lie algebra. Remark: After proving that g is isomorphic to the vec- tor space of left-invariant vector fields on G,wegetan- other proof of Proposition 16.2. 16.2. LEFT AND RIGHT INVARIANT VECTOR FIELDS, EXPONENTIAL MAP 775 16.2 Left and Right Invariant Vector Fields, the Ex- ponential Map Afairlyconvenientwaytodefinetheexponentialmapis to use left-invariant vector fields. Definition 16.7. If G is a Lie group, a vector field, X, on G is left-invariant (resp. right-invariant)i↵ d(L ) (X(b)) = X(L (b)) = X(ab), for all a, b G. a b a 2 (resp. d(R ) (X(b)) = X(R (b)) = X(ba), for all a, b G.) a b a 2 Equivalently, a vector field, X,isleft-invarianti↵the following diagram commutes (and similarly for a right- invariant vector field): d(La)b T G / T G b O abO X X G / G La 776 CHAPTER 16. LIE GROUPS, LIE ALGEBRA, EXPONENTIAL MAP If X is a left-invariant vector field, setting b =1,wesee that X(a)=d(La)1(X(1)), which shows that X is determined by its value, X(1) g, at the identity (and similarly for right-invariant vector2 fields). Conversely, given any v g,wecandefinethevector field, vL,by 2 vL(a)=d(L ) (v), for all a G. a 1 2 We claim that vL is left-invariant. This follows by an easy application of the chain rule: 16.2. LEFT AND RIGHT INVARIANT VECTOR FIELDS, EXPONENTIAL MAP 777 L v (ab)=d(Lab)1(v) = d(L L ) (v) a ◦ b 1 = d(La)b(d(Lb)1(v)) L = d(La)b(v (b)). Furthermore, vL(1) = v. In summary, we proved the following result. Proposition 16.3. Given a Lie group G, the map X X(1) establishes an isomorphism between the 7! space of left-invariant vector fields on G and g.In fact, the map G g TG given by (a, v) vL(a) is ⇥ ! 7! an isomorphism between G g and the tangent bundle TG. ⇥ Definition 16.8. The vector space of left-invariant vec- tor fields on a Lie group G is denoted by gL. 778 CHAPTER 16. LIE GROUPS, LIE ALGEBRA, EXPONENTIAL MAP Because the derivative of any Lie group homomorphism is a Lie algebra homomorphism, (dLa)b is a Lie algebra homomorphism, so gL is a Lie algebra.