Chapter 7 - Lie Groups

Question 1

Define: Lie group, left/right translation

Answer 1

A LIE GROUP is a smooth manifold G (without boundary) that is also a group in the algebraic sense, with the property that the multiplication map m: G x G –> G and the inversion map i : G –> G, given by
m(g,h) = gh anfd i(g) = g^-1
are both smooth.

Any element g of G defines maps L_g, R_g : G –> G called LEFT TRANSLATION and RIGHT TRANSLATION respectively, by
L_g(h) = gh and R_g(h) = hg

L_g is composition of smooth maps m o i_g where i_g(h) = (g,h) so smooth. In fact, L_g is a diffeomorphism since L_g^-1 is inverse map. Same for R_g.

KEY: Many of the important properties of Lie groups come from the fact that we can systematically map any point to any other by such a global diffeomorphism

Question 2

Key property of Lie groups?

Answer 2

KEY: Many of the important properties of Lie groups come from the fact that we can systematically map any point to any other by global diffeomorphisms L_g or R_g

Question 3

Examples of Lie groups?

Answer 3

1. GL(n, R), GL(n, C), GL(V) where V is a real or complex vector space. If f.d. isomorphic to GL(n, k)
2. GL^+(n,R)

151-152

Question 4

Define: Lie group homomorphism, isomorphism, isomorphic Lie groups

Examples?

Answer 4

If G and H are Lie groups, a LIE GROUP HOMOMORPHISM from G to H is a smooth map F: G –> H that is also a group homomorphism. It is called a LIE GROUP ISOMORPHISM if it is also a diffeomorphism, which implies that is has an inverse that is also a Lie group homomorphism. In this case say G and H are ISOMORPHIC LIE GROUPS.

Examples

1. Inclusion S^1 –> C*
2. exp: R –> R, exp(t) = e^t. exp: R –> R^+ is an isomorphism. exp: C –> C surjective, not injective
3. epsilon: R –> S^1, epsilon(t) = e^2piit, ker = Z. Similarly epsilon^n:R^n –> T^n ker = Z^n
4. det : GL(n,k) –> k*
5. Conjugation by g in G, C_g: G –> G, C_g(h) = ghg^-1

Question 5

Key property of Lie group homomorphisms? Proof? Corollary?

Answer 5

(compare to equivariant rank theorem)
Thm. Every Lie group homomorphism has constant rank

Pf. Let F: G –> H be a Lie group hom and e , e’ denote the identity elements of G, H. Suppose g0 is arbitrary. We show dF_g0 has the same rank as dF_e. Since F is a homomorphism, for all g in G:
F(L_g0(g)) = F(g0g) = F(g0)F(g) = L_F(g0)(F(g))
i.e. F o L_g0 = L_F(g0) o F. Taking differentials at identity ans using “chain rule”, we find:
dF_g0 o d(L_g0)_e = d(L_F(g0))_e’ o dF_e.

Now L_g is a diffeo, so d(L_g0)_e and d(L_F(g0))_e’ are isomorphisms. Since composing with isomorphism does not change rank of map, it follows that dF_g0 and dF_e have the same rank.

Corollary. A Lie group homomorphism is a Lie group isomorphism <=> it is bijective.

Pf. Apply global rank thm.

Question 6

What can be said about covering groups of Lie groups?
Proof?

Examples

Answer 6

Thm. Let G be a connected Lie group. There exists a simply connected Lie group G tilda, called the UNIVERSAL COVERING GROUP OF G, that admits a smooth covering map pi : G tilda –> G that is also a Lie group homomorphism. G tilda is unique up to isomorphism.

Pf. We have a universal covering manifold G tilda of G and smooth covering map pi. Choose any element e tilda in pi^-1(e). Since G tilda simply connected, the lifting criterion for covering maps guarantees that the map m o (pi x pi) : G tilda x G tilda –> G has a unique continous lift m tilda.

Show m tilda smooth. Do same for i.

All that is left is to shpw G tilda is a group with these operations. This all comes down to the uniqueness of lifts.

Examples

1. epsilon^n : R^n –> T^n
2. exp: C –> C*

pg 154 -155

Question 7

Define: Lie subgroup of G, identity component of G

Answer 7

A LIE SUBGROUP OF G is a subgroup of G endowed with a topology and smooth structure making it into a Lie group and an immersed submanifold of G

The connected component of G containing the identity is called the IDENTITY COMPONENT OF G

Question 8

Show embedded subgroups are automatically Lie subgroups

Answer 8

Say H is an embedded subgroup. Then H is already endowed with topology and smooth structure making it into embedded (so immersed) submanifold of G. We need only show the group operations when restricted to H are still smooth. Consider multiplication. This is smooth from G x G –> G, so smooth from H x H –> G (even if H is just immersed). Since H is a subgroup, multiplication takes H into H and since H is embedded, this is a smooth map (remember smooth embeddings behave well with restriction of both domain and CODOMAIN). Same for inversion.

Question 9

What do the open subgroups of Lie groups look like?

Answer 9

Open subgroups are automatically embedded so automatically Lie subgroups. It turns out that they are always a union of connected components of G.

Every left coset gH is open in G because it is the image of the open subset H under the diffeomorphism L_g. Cosets partition G. Thus G \ H = union of cosets of H other than H, is open, therefore H is closed in G. Since H both open and closed, it is a union of components.

Question 10

Discuss (with proof) the group generated by an arbitrarily small neighborhood W of the identity in G a Lie group

Answer 10

1. W generates an open subgroup of G
2. W is connected
3. If G is connected, then W generates G

So for connected Lie groups, all info about the group is contained in an arbitrarily small neighborhood of the identity.

Pf. (1) Let H be the subgroup generated by W. Let W_k denote the set of elements of G that can be expressed as a product of k or fewer elements of W U W^-1. Notice H = U_k=1^inf W_k. We show H is open inductively.

W_1 = W U W^-1 is open since W open and W^-1 image of open under inversion map which is a diffeomorphism. Now W_k = W_1W_k-1 = U_g in W_1 L_g(W_k-1)
Because each L_g is a diffeomorphism, it follows by induction that each W_k is open, and thus H is open.

Next, if W connected, then W_1 connected since it is W U W^-1. W^-1 connected being image of W under diffeo and both W and W^-1 contain e. Therefore W_2 = m(W1 x W1) is connected, being the image of a connected space under the continuous multiplication map, so W_k is connected by induction and H = union W_k is connected being union of connected subsets with idenity in common.

Finally, if G is connected, we notice H is open and closed and contains e so nonempty. Thus H = G.

Question 11

Discuss the kernel and image of a Lie group homomorphism F : G –> H

Answer 11

Prop. The kernel of F is a properly embedded Lie subgroup of G, whose codimension is equal to the rank of F.

Pf. Because F has constant rank, its kernel F^-1(e) is a properly embedded submanifold of codimension equal to rank F (Constant rank level set thm). We saw above that embedded subgroups are automatically Lie subgroups (came down to being able to restrict codomain of embedded submanifold).

Now images of F are also are Lie subgroups. For now, we only have tools to prove:

Prop. If F: G –> H is an injective Lie group homomorphism, the image of F has a unique smooth manifold structure such that F(G) is a Lie subgroup of H and F : G –> F(G) is a Lie group isomorphism.

Pf. Since Lie group hom has constant rank, we get that F is a smooth immersion from the global rank thm. Recall immersed submanifolds = images of injective smooth immersions. So F(G) has a unique smooth manifold structure such that it is an immersed submanifold of H and F is a diffeomorphism onto its image. It is a Lie group (because G is) and it is a subgroup for algebraic reasons, so it is a Lie subgroup. Because F: G –> F(G) is a group iso and a diffeo, it is a Lie group iso.

In CH 21, we prove the more general result:

Thm. (First Isomorphism Thm for Lie Groups). If F: G –> H is a Lie group homomorphism, then kernel of F is a closed normal subgroup of G, the image of F has a unique smooth manifold structure making it into a Lie subgroup of H, and F descends to a Lie group isomorphism F tilda : G / Ker F –> Im F. If F is surjective, then G / Ker F is smoothly isomorphic to H.

Question 12

Examples of Lie Subgroups?

Answer 12

Embedded

1. GL^+(n, R) < GL(n, R)
2. S^1 < C*
3. SL(n, R), ker of Lie group hom det : GL(n,R) –> R*
4. GL(n, C) < GL(2n, R)
5. SL(n, C) < GL(n,C) < GL(2n, R)

Not Embedded
Let H < T^2 be the dense submanifold of the torus that is the image of the immersion gamma: R –> T^2. This is an injective Lie group homomorphism, so H is an immersed Lie subgroup of T^2

Question 13

What is the relationship between closed as a subspace and embedded as a submanifold? What about for Lie subgroups?

Answer 13

For general smooth submanifolds:

1. Closed need not be embedded – figure 8 curve
2. Embedded need not be closed – open unit ball in R^n

For Lie subgroups:
A subgroup H of G is closed in G <=> it is embedded.

In Ch. 20 prove a stronger version of this: CLOSED SUBGROUP THEOREM every subgroup of a Lie group that is topologically closed subset (not assumed to be a submanifold) is automatically a properly embedded Lie subgroup.

Question 14

Define: left action of G on M, right action, continuous action, G-space, smooth action

Importance?

Answer 14

If G is a group and M is a set, a LEFT ACTION OF G ON M is a map G x M –> M, (g,p) |–> gp that satisfies
g1(g2p) = (g1g2)p
ep = p

If M is a topological space and G is a topological group, an action of G on M is s.t.b a CONTINUOUS ACTION if the defining map G x M –> M is continuous. In this case, M is said to be a G-SPACE. If in addition M is a smooth manifold, G is a Lie group, and the defining map is smooth, then the action is s.t.b. a SMOOTH ACTION.

The most important applications of Lie groups to smooth manifold theory involve actions by Lie groups on other manifolds. If M is a smooth manifold endowed with a metric or other geometric structure, the set of diffeomorphisms of M that preserve the structure (called the SYMMETRY GROUP of the structure) frequently turns out to be a Lie group acting smoothly on M.

Question 15

Define: orbit of p, isotropy group, transitive action, free action

Answer 15

For each p in M, the ORBIT of p, denoted G dot p, is the set of all images of p under the action by elements of G: G dot p = { gp : g in G}

For each p in M, the ISOTROPY GROUP or STABILIZER OF P, denoted G_p is the set of elements of G that fix p:
G_p = {g in G : gp = p}

The action is TRANSITIVE if for every pair of points p, q in M, there exists g in G s.t. gp = q, or equivalently the only orbit is all of M

The action is FREE if the only element of G that fixes any element of M is the identity: gp = p for some p implies g = e, or equivalently if every isotropy group is trivial

Question 16

Examples of Lie Group Actions?

Answer 16

1. If G is any Lie group and M is any smooth manifold, the TRIVIAL ACTION OF G ON M defined by gp = p. Orbits are points. Isotropy groups are all of G
2. Natural action of GL(n,R) on R^n: Ax. Two orbits {0} and R^n \ {0}
3. Every Lie group G acts smoothly on itself by LEFT TRANSLATION G, defines a smooth and free (but generally not transitive) left action of H on G
4. Every Lie group acts smoothly on itself by CONJUGATION
5. An action of a discrete group Gamma on M is smooth <=> for each g in Gamma, the map p –> gp is a smooth map from M to itself
6. Suppose E and M are smooth manifolds and pi : E –> M is a smooth covering map. With the discrete topology, the automorphism group Aut_pi(E) is a zero-dimensional Lie group acting smoothly and freely on E.

It acts transitively on each fiber of pi <=> pi is a normal covering map.

Question 17

Define: equivariant

Answer 17

Suppose G is a Lie group, and M and N are both smooth manifolds endowed with smooth left or right G-actions. A map F:M —> N is said to be equivariant with respect to the given G-actions if for each g in G,
F(gp) = gF(p) (for left actions);

Also say F intertwines

Question 18

Examples of equivariant maps?

Answer 18

Consider the action of R on R^n and R on T^n by:
t(x1, … , xn) = (x1 + tv1, … , x^n _+tvn)
t(z1, … , zn) = (e^2piitv1 z1, … , e^2piitvn zn)
The smooth map epsilon^n : R^n –> T^n is equivariant w.r.t. these actions pg 164 -165

Question 19

What is Equivariant Rank Theorem? Proof?

Answer 19

Thm. Let M, N be smooth manifolds and G be a Lie group. Suppose F:M –> N is a smooth map that is equivariant with respect to a TRANSITIVE smooth G-action on M and ANY smooth G-action on N. Then F has constant rank.

Cor. By Global Rank Thm,
F surjective –> smooth submersion
F injective –> smooth immersion
F bijective –> diffeomorphism

Pf. Let theta and phi denote the G-actions on M and N and let p, q be arbitrary points in M. Choose g in G s.t. theta_g(p) = q (such a g exists bevcause we are assuming G acts transitively on M). By equivariance, phi_g o F = F theta_g. Take differentials and not dtheta_g and dphi_g are isomorphisms so dF_p and dF_q have the same rank.

Question 20

Discuss the orbit map and its properties.

Answer 20

G Lie group, M smooth manifold acted on by G. For each p in M, define a map theta^(p) : G –> M by theta^(p)(g) - gp. This is called the ORBIT MAP, its image is the orbit G dot p. Its preimage (theta^(p))^-1(p) is the isotropy group G_p.

Properties: theta^(p) is smooth and has constant rank, so the isotropy group G_p is a properly embedded Lie subgroup of G. If G_p = {e}, then theta^(p) is an injective smooth immersion, so the orbit G dot p is an immersed submanifold of M. (in fact, in CH 21 we see that every orbit is an immersed submanifold of M)

Pf. Oribit map is smooth because it is composition of G x {p} –> G x M –> M. It is equivariant by def of left translation action. Since G acts transitively on itself, the equivariant rank thm shows theta^(p) has constant rank. Thus G_p is a properly embedded submanifold by the Constant Rank Level Set Theorem and a Lie subgroup by an above result.

If G_p = {e}, theta^(p) is injective. By equivariant rank thm, it is smooth immersion, thus orbit is an immersed submanifold.

Question 21

Use the equivariant rank theorem to identify important Lie subgroups of the general linear groups

Answer 21

Recall: a linear map is ORTHOGONAL if it preserves the Euclidean dot product (Ax) dot (Ay) = x dot y. The set O(n) of all orthogonal nxn ,matrices is a subgroup of GL(n,R) called the ORTHOGONAL GROUP of degree n. Define a smooth map phi: GL(n, R) –> M(n,R) by phi(A) = A^TA. Then O(n) is equal to the level set phi^-1(I_n). To show phi has constant rank, and therefor that O(n) is an embedded Lie subgroup, we show that phi is equivariant with respect to suitable right actions of GL(n,R). Let GL(n,R) act on itself by right multiplication and on M(n,R) by X dot B = B^T XB for X in M(n,R), B in GL(n,R). Check this is equivariant. Thus, O(n) is a properly embedded Lie subgroup of GL(n,R). It is compact because it is closed (level set) and bounded (columns of norm 1). To determine the dimension, we just need to compute the rank of phi – dim O(n) = n(n-1)/2

SO(n)? U(n)? SU(n)?

Question 22

Discuss Semidirect Products of Lie Groups. Examples? Properties?

Answer 22

Suppose H and N are Lie groups and N has an H-action by automorphisms - i.e. theta_h : N –> N is an automorphism for all h in H. Then we can define a new Lie group N semidirect H called the SEMIDIRECT PRODUCT OF H AND N as follows. As a smooth manifold, N semidirect H is just cartesian product N x H, but group multiplication is defined by (n, h)(n’, h’) = (n(hn’), hh’).

Examples
Euclidean Group: O(n) acts on R^n by automorphisms. The resulting semidirect product E(n) = R^n semidirect O(n) is the EUCLIDEAN GROUP. Multiplication (b, A)(b’, A’) = (b + Ab’, AA’).

It acts on R^n by (b,A)x = b + Ax nh is a Lie group isomorphism between N semidirect H and G.

Question 23

Define: f.d. representation of G, faithful, defining representation. Examples? Relation to linear actions?

Answer 23

If G is a Lie group, a f.d. REPRESENTATION OF G is a Lie group homomorphism from G to GL(V) for some V. If rho: G –> GL(V) is injective, it is stb FAITHFUL. In that case the image of rho is a Lie subgroup of GL(V) . Thus a Lie group admits a faithful representation <=> it is isomorphic to a Lie subgroup of GL(n,R) or GL(n,C) for some n. Not every Lie group admits such a representation - universal cover of SL(2,R).

Examples
1. If G is any Lie subgroup of GL(n,R), the inclusion map G –> GL(n,R) is a faithful representation called the DEFINING REPRESENTATION OF G.

2. S^1 –> C* or T^n –> GL(n,C) is faithful rep of T^n
3. R^n –> GL(n+1, R) faithful representation of additive Lie group
4. R^n –> GL(n,R) (x1, … , xn) –> (e^x1, … , e^xn)
5. R^n –> GL(n, C) (x1, … , xn) –> (e^2piix1, … , e^2piI*xn)
6. GL(n, R) –> GL(P^n_d) polynomial functions of degree at most d

A smooth left action of G on V is linear <=> it is of the form g dot x = rho(g)x for some representation rho of G.