Riemannian Geometry – Lecture 17

Home , Diffeomorphism, General linear group, Homogeneous space, Linear group, Special unitary group, Unitary group

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) = {A ∈ Mn(R) | det A 6= 0}

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) = {A ∈ Mn(R) | det A > 0} 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) = {A ∈ Mn(R) | A = A} 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 ∈ R . Example 17.9 The orthogonal group is deﬁned by

t O(n) = {G ∈ Mn(R) | AA = I}.

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 ∈ O(n) ⇔ hAx, Ayi = hx, yi for all x, y ∈ R . Example 17.11 Consider the special orthogonal group

t SO(n) = {A ∈ Mn(R) | det A = 1 and AA = I}.

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 ∈ 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) ∈ W,

 1 1 1  F1 (a, b) F2 (a, b) ··· Fn (a, b)  F 2(a, b) F 2(a, b) ··· F 2(a, b)   1 2 n   . . . .   . . . .  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) ∈ 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 ∈ U(n) ⇔ (Az, Aw) = (z, w) for all z, w ∈ C . The space of Hermitian symmetric matrices t HermSym(n) = {A ∈ Mn(C) | A = A} consists of matrices of the form   x1 z12 z13 ··· z1n  z x z ··· z   12 2 23 2n   . . .   z z .. . .   13 23   . . .   . . . xn−1 zn−1,n    . . z1n . . zn−1,n xn where zij ∈ C, xi ∈ R and hence is a manifold diffeomorphic to ( − )+ 2 Rn n 1 n = Rn . Exercise 17.18 Show that A ∈ HermSym if and only if for all z, w ∈ 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 ∈ 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 ∈ M and

(gh) · p = g · (h · p) for all g, h ∈ G, p ∈ M.

(these force the action of each element to be a bijection) Deﬁnition 17.21 (continued) if for every g ∈ 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 ∈ M there exists g ∈ 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 ∈ 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.