Riemannian Geometry – Lecture 17 Lie Groups Dr. Emma Carberry September 21, 2015 Lie groups Definition 17.1 A Lie group is a manifold which is also a group and is such that the group operations of 1 group multiplication G × G ! G :(g; h) 7! gh and 2 taking inverses G × G : g 7! g−1 are both smooth. Example 17.2 Rn is a Lie group under addition. Lie groups Let Mn(R) denote the space of n × n matrices with real entries. Example 17.3 The general linear group GL(n; R) = fA 2 Mn(R) j det A 6= 0g 2 is an open subspace of Rn and hence a manifold of dimension n2. It is easy to verify that the group operations (g; h) 7! gh and g 7! g−1 are smooth. More intrinsically, GL(n; R) is the space of invertible linear transformations Rn ! Rn. Example 17.4 The positive general linear group + GL (n; R) = fA 2 Mn(R) j det A > 0g is a Lie subgroup (i.e. a subgroup and a submanifold) of the general linear group and has the same dimension n2 since it is an open subset. Exercise 17.5 + GL (n; R) is the group of linear transformations Rn ! Rn which preserve orientation. Recall: Definition 17.6 An orientation of an n-dimensional vector space V is a choice of one of the two possible equivalence classes of ordered basis for V under the equivalence relation ∼ defined by: (X1;:::; Xn) ∼ (Y1;:::; Yn) for ordered bases (X1;:::; Xn) and (Y1;:::; Yn) of V if the n × n matrix A defined by (Y1;:::; Yn) = (X1;:::; Xn)A has positive determinant. Example 17.7 The set of symmetric matrices t Sym(n) = fA 2 Mn(R) j A = Ag is not a Lie group under multiplication since it contains the zero matrix, and so is not a group. It is however a manifold since taking the diagonal and upper triangular entries defines a global diffeomorphism to Rk , where n(n + 1) k = 1 + 2 + ··· + n = : 2 Exercise 17.8 A matrix is symmetric if and only if n hAx; yi = hx; Ayi for all x; y 2 R : Example 17.9 The orthogonal group is defined by t O(n) = fG 2 Mn(R) j AA = Ig: Note that this forces the determinant of A to be either 1 or −1. In fact the group O(n) consists of two connected components. Exercise 17.10 Prove that O(n) is the group of linear transformations Rn ! Rn which preserve the Euclidean inner product in the sense that n A 2 O(n) , hAx; Ayi = hx; yi for all x; y 2 R : Example 17.11 Consider the special orthogonal group t SO(n) = fA 2 Mn(R) j det A = 1 and AA = Ig: Exercise 17.12 Prove that SO(n) is precisely the group of linear transformations Rn ! Rn which preserve the Euclidean inner product and orientation. Example 17.11 (Continued) Moreover defining + f : GL (n; R) ! Sym(n) A 7! t AA − I; we have SO(n) = f −1(0). The map f is smooth, and furthermore it is a submersion. Definition 17.13 A smooth map f : Mn+m ! Nn between smooth manifolds is a submersion at p 2 M if dfp has (maximal) rank n. The map is a submersion if it is is a submersion at every point in its domain. Equivalently, dfp is surjective. Example 17.11 (Continued) In our example we have + n2 ∼ n(n+1)=2 f : GL (n; R) ⊂ R ! Sym(n) = R open and f being a submersion tells us that it satisfies the criteria of the implicit function theorem. Recall Theorem 17.14 (Implicit Function Theorem) Suppose + F : W ⊂ Rm n = Rm × Rn ! Rn 1 n (x1;:::; xm; y1;:::; yn) 7! (F (x; y);:::; F (x; y)) is a smooth map, and that for (a; b) 2 W, 0 1 1 1 1 F1 (a; b) F2 (a; b) ··· Fn (a; b) B F 2(a; b) F 2(a; b) ··· F 2(a; b) C B 1 2 n C B . C @ . A n n n F1 (a; b) F2 (a; b) ··· Fn (a; b) i is invertible, where F i = @F . Write c = F(a; b). Then there are j @yj open neighbourhoods U ⊂ Rm of a and V ⊂ Rn of b and a smooth map g : U ! V so that for (x; y) 2 U × V, F(x; y) = c , y = g(x): Example 17.11 (Continued) The implicit function theorem tells us then that SO(n) = f −1(0) is a manifold of dimension n(n + 1) 2n2 − n2 − n n(n − 1) n2 − = = : 2 2 2 The group operations (multiplication and taking inverses) are smooth and hence SO(n) is a Lie group. Example 17.15 Similarly, applying the same argument to f : GL(n; R) ! Sym(n) A 7! t AA − I; we recognise O(n) as f −1(0) and hence see that O(n) is a Lie n(n−1) group of dimension 2 . As stated above, O(n) has two connected components, one of which is SO(n). Example 17.16 The unitary group U(n) ⊂ GL(n; C) is the subgroup of the complex general linear group consisting of matrices satisfying t AA¯ = I. The special unitary group SU(n) consists of those unitary matrices which additionally satisfy det A = 1. Exercise 17.17 Prove that U(n) is the group of linear transformations Cn ! Cn which preserve the Hermitian inner product (z; w) = z1w1 + ··· znwn in the sense that n A 2 U(n) , (Az; Aw) = (z; w) for all z; w 2 C : The space of Hermitian symmetric matrices t HermSym(n) = fA 2 Mn(C) j A = Ag consists of matrices of the form 0 1 x1 z12 z13 ··· z1n B z x z ··· z C B 12 2 23 2n C B . C B z z .. C B 13 23 C B . C B . xn−1 zn−1;n C @ A . z1n . zn−1;n xn where zij 2 C; xi 2 R and hence is a manifold diffeomorphic to ( − )+ 2 Rn n 1 n = Rn . Exercise 17.18 Show that A 2 HermSym if and only if for all z; w 2 Cn, (Az; w) = (z; Aw): Active Learning Question 17.19 By an analogous argument to that used for SO(n), prove that U(n) is a real smooth manifold of dimension n2. Note that a unitary matrix satisfies det A det A = 1 and hence det A 2 S1. We may characterise SU(n) as the pre-image of 1 under the map det : U(n) ! S1: Again det is a submersion and so the implicit function theorem tells us that SU(n) is a smooth real manifold of dimension n2 − 1. Definition 17.21 a Lie group G acts on a manifold M if there is a map G × M ! M (g; p) 7! g · p such that e · p = p for all p 2 M and (gh) · p = g · (h · p) for all g; h 2 G; p 2 M: (these force the action of each element to be a bijection) Definition 17.21 (continued) if for every g 2 G, g : M ! M p 7! g · p is smooth then we say that G acts smoothly the isotropy subgroup Gp of G at p is the subgroup fixing p G acts transitively on M if for every p; q 2 M there exists g 2 G such that q = g · p Definition 17.22 A homogeneous space is a manifold M together with a smooth transitive action by a Lie group G. Informally, a homogeneous space “looks the same” at every point. Since the action is transitive, the isotropy groups are all conjugate −1 Gg·p = gGpg and for any p 2 M we can identify the points of M with the quotient space G=Gp. Definition 17.23 A Lie subgroup H of a Lie group G is a subset of G such that the natural inclusion is an immersion and a group homomorphism (i.e. H is simultaneously a subgroup and submanifold). Since the the group action on a homogeneous space is in particular continuous, the isotropy subgroups are closed in the topology of G. Theorem 17.24 (Lie-Cartan) A closed subgroup of a Lie group G is a Lie subgroup of G. Corollary 17.25 An isotropy subgroup Gp of a Lie group G is a Lie subgroup of G. Theorem 17.26 If G is a Lie group and H a Lie subgroup then the quotient space G=H has a unique smooth structure such that the map G × G=H ! G=H (g; kH) 7! gkH is smooth. We omit the proof; it is given for example in Warner, “Foundations of Differentiable Manifolds and Lie Groups”, pp 120–124. Corollary 17.27 A homogeneous space M acted upon by the Lie group G with isotropy subgroup Gp is diffeomorphic to the quotient manifold G=Gp where the latter is given the unique smooth structure of the previous theorem..