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Appendix B Lie Groups and Lie Algebras

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Appendix B Lie Groups and Lie Algebras 196 APPENDIX B. LIE GROUPS AND LIE ALGEBRAS this notion is already familiar to the reader. From a geometric perspective, it becomes more interesting if we assume that G is also a topological space or a manifold. Definition B.1. A topological group G is a group with a topology such Appendix B that the maps G G G : (a; b) ab and G G : a a−1 × ! 7! ! 7! are both continuous. Lie groups and Lie algebras Similarly, a Lie group is a group that is also a smooth manifold, such that the two maps above are both smooth. A Lie subgroup is a subgroup H G that is also a submanifold, i.e. the inclusion map H , G is both a group⊂ homomorphism and a smooth embedding. ! Contents The groups that are of greatest interest in geometry and topology typ- ically consist of bijective maps ' : M M on some space M, with the B.1 Topological and Lie groups . 195 ! group multiplication law given by composition, B.2 Lie algebras in general . 206 ' := ' ; B.3 Lie algebras of matrix groups . 207 ◦ B.4 General Lie groups and their Lie algebras . 211 and inversion being the obvious operation ' '−1. If M is a topological space, then there are various possible topologies7! one can define on spaces of continuous maps M M, depending sometimes on extra (e.g. smooth) structures; these details!will not be integral to our discussion, so we will B.1 Topological and Lie groups suppress them. It will at least be clear in all the examples of interest that the groups under consideration can be regarded as topological groups. The simplest example is Almost every interesting bundle comes with some extra structure attached to its fibers, for instance the vector space structure for a vector bundle, Homeo(M) = ' : M M ' is bijective, ' and '−1 both continuous ; or an inner product (bundle metric) or orientation. One of the useful f ! j g things about the concept of a bundle is that it allows us to treat all of where M is any topological space. If M is also an orientable topological these structures as special cases of the same thing: in principle one need manifold, then Homeo(M) has an important subgroup only specify the structure group of the bundle. This notion is explained Homeo+(M) = ' Homeo(M) ' preserves orientation ; in Chapter 2, and plays an important role in the theory of connections in f 2 j g Chapter 3. Since one must first understand the basic concepts of Lie groups we refer to [Hat02] for the definition of \orientation preserving" on a topo- and their associated Lie algebras, we shall now give a quick overview of logical manifold. If M is a smooth manifold, there are the corresponding these ideas. Besides, Lie groups are interesting in their own right. subgroups Recall first that a group is a set G together with an operation G G × ! G, usually denoted as \multiplication" (a; b) ab.1 The operation is Diff(M) = ' Homeo(M) ' and '−1 are smooth ; 7! required to be associative, a(bc) = (ab)c, and there is a special element f 2 j g e G, called the identity, such that eg = ge = g for all g G. Moreover, and in the orientable case, ev2ery g G has an inverse g−1 G such that g−1g = gg−1 2= e. We expect 2 2 Diff+(M) = ' Diff(M) ' preserves orientation : f 2 j g 1Sometimes the group operation is denoted alternatively as \addition" (a; b) a+b. This is the custom particularly when the operation is commutative, i.e. a + b =7!b + a, There are still other groups corresponding to extra structures on a smooth in which case we say G is abelian. manifold M, such as a Riemannian metric g or a symplectic form !. These 195 B.1. TOPOLOGICAL AND LIE GROUPS 197 198 APPENDIX B. LIE GROUPS AND LIE ALGEBRAS two in particular give rise to the groups of isometries and symplectomor- det( + B) is a continuous function of B. Defining a norm B on matrices 2 j j phisms respectively, by identifying Fn×n with Fn , it's not hard to show that there is a constant AB A B Bk k−1 B k ∗ C > 0 such that C , and in particular C = Isom(M; g) = ' Diff(M) ' g g ; 1 (C B )k. Thus bj y ajstandard≤ j jj argumenj t from analysis,j thej ≤ infinitej jsum f 2 j ∗ ≡ g C j j Symp(M; !) = ' Diff(M) ' ! ! : above can be bounded by f 2 j ≡ g Most of these examples are infinite dimensional,2 which introduces some 1 1 1 1 complications from an analytical point of view. They cannot be considered ( 1)kBk Bk (C B )k < − ≤ j j ≤ C j j 1 smooth Lie groups, though they clearly are topological groups. k=0 k=0 k=0 X X X Actual Lie groups are usually obtained by considering linear transfor- F whenever B is small enough so that C B < 1. Now one sees by an easy mations on finite dimensional vector spaces. As always, let denote either j j R or C. Groups which are smooth submanifolds of the vector space computation that n×n 1 1 F := all n-by-n matrices with entries in F f g ( + B) ( 1)kBk = ( 1)kBk ( + B) = : − − k=0 k=0 ! are called linear groups or matrix groups. Denote by n (or simply when X X there's no ambiguity) the n-by-n identity matrix We apply this as follows to show that the map ι : GL(n; F) Fn×n : A −1 H Fn×n ! 7! 1 0 0 A is smooth: for sufficiently small, we have · · · 2 0 1 0 0 · · · 1 ι(A + H) = (A + H)−1 = ( + A−1H)−1A−1 = n = . .. ; . B C = A−1H + (A−1H)2 : : : A−1 B0 0 1C − − B · · · C = ι(A) A−1HA−1 + ( H 2); @ A − O j j this will be the identity element of every linear group. Now observe that the map proving that ι is differentiable and dι(A)H = A−1HA−1. (Notice that Fn×n Fn×n Fn×n d 1 1 − : (A; B) AB this generalizes the formula dx x = x2 .) Next observe that the map × ! 7! GL(n; F) Hom(Fn×n; Fn×n) : A −dι(A) is a quadratic function of the is smooth; indeed, it's a bilinear map, and every bilinear map on a finite ! 7! differentiable map A A−1, and is thus also differentiable. Repeating this dimensional vector space is smooth (prove it!). A slightly less obvious but argument inductively,7!we conclude that ι has infinitely many derivatives. very important fact is that if we define the open subset This fact lays the groundwork for the following examples. GL(n; F) = A Fn×n det(A) = 0 ; f 2 j 6 g Example B.2. The set GL(n; F) defined above is precisely the set of all then the map GL(n; F) Fn×n : A A−1 is also smooth. This is invertible n-by-n matrices: this is called the general linear group. As an easy to see in the case n != 2 since you can7! write down A−1 explicitly; in open subset of the vector space Fn×n, it is trivially also a smooth manifold fact it follows more generally from Cramer's rule, an explicit formula for (of dimension n2 if F = R, or 2n2 if F = C), and the remarks above inverses of arbitrary invertible matrices, but that's not the clever way to show that multiplication and inversion are smooth, so that GL(n; F) is do it. The clever way is to observe first that the power series expansion a Lie group. It's useful to think of GL(n; F) as the set of invertible F- 1 2 3 A Fn Fn 1+x = 1 x + x x + : : : can be generalized to matrices: we have linear transformations : , with group multiplication defined − − by composition of transformations.! In this way GL(n; F) is naturally a ( + B)−1 = B + B2 B3 + : : : (B.1) subgroup of Diff(Fn). The next several examples will all be Lie subgroups − − R C of either GL(n; ) or GL(n; ). for any matrix B sufficiently close to 0. Indeed, we see that ( + B) C R In fact, one can also regard GL(n; ) as a Lie subgroup of GL(2n; ). B is necessarily invertible if is sufficiently small, since det( ) = 1 and Indeed, if we identify Cn with R2n via the bijection 2The exception is Isom(M; g), which is generated infinitessimally by the finite di- mensional space of Killing vector fields; see e.g. [GHL04]. R2n Cn : (x1; : : : ; xn; y1; : : : ; yn) (x1 + iy1; : : : ; xn + iyn); ! 7! B.1. TOPOLOGICAL AND LIE GROUPS 199 200 APPENDIX B. LIE GROUPS AND LIE ALGEBRAS n then multiplication of i by vectors in C becomes a real linear transforma- where the vectors vj form the columns of an n-by-n matrix. Thus GL+(n; R) 2n 2n n tion J0 : R R represented by the matrix is precisely the set of orientation preserving linear transformations R ! Rn ! . 0 n J = ; 0 −0 Example B.4. The special linear group is defined by n and multiplication of a complex scalar a + ib by vectors in R2n is now the SL(n; F) = A Fn×n det(A) = 1 : 2n×2n n×n f 2 j g linear transformation a +bJ0 R . Any A C defines a complex linear transformation Cn Cn2, and therefore also2a real linear transforma- This is obviously a subgroup, and we claim that it is also a smooth sub- 2 2 R2n R2n ! manifold of GL(n; F), with dimension n 1 in the real case, 2n 2 in tion , but with the extra property of preserving complex scalar − − multiplication:! this means that for all v R2n and a + ib C, the complex case.
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