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1 1 Lie Groups Definition (4.1 1 A Lie Group G is a set that is a group a differential manifold with the property that and are smooth. µ : G G G (g 1, g 2 g 1 g 2 i : G G g g 1 Definition (4.1 2 A Lie Subgroup of G is a subset H of G such that (i H is a subgroup of G and (ii H is a submanifold of G and (iii topological group with respect to subspace topology. Definition (4.1 3 The left and right translations are diffeomorphisms of G defined by l g and r g satisfy r g : G G, l g : G G g g g g gg. l g1 l g2 = l g1 g 2 and r g1 r g2 = r g2 g 1. g l g and g r g are an isomorphism and an anti-isomorphism (bijection, φ(ab = φ(bφ(a and same for inverse, respectively. A homomorphism of Lie groups is a smooth group homomorphism. 1.1 Examples 1. R n with + 2. S 1 := {x C x = 1}: Circle in complex plane is group under multiplication but also manifold (circle. 1
2 3. real general linear group GL(n, R := {A M(n, R det A }. Differential structure given by bijection with R n2. Because det is continuous and {} is closed det 1 ( is closed and the complement, GL(n, R is open. Every open subset of an n-dimensional manifold is an n-dimensional submanifold. Decomposes into two disjoint components with det > / < Dimension is n Similarly GL(n, C := {A M(n, C det A }, dimension 2n 2. But GL(n, C is connected while GL(n, R is not. 5. connected component of GL(n, R: GL + (n, R := {A M(n, R det A > } this is subgroup of GL(n, R because 1 GL + (n, R det AB = det A det B (A, B GL + (n, R AB GL + (n, R det A 1 = (det A 1 (A GL + (n, R A 1 GL + (n, R 1 GL + (n, R 6. SL(n, R := {A M(n, R det A = 1} 7. O(n, R := {A GL(n, R AA T = 1} det A = ±1 is a compact Lie Group of (n 2 n/2 dimensions. 8. SO(n, R := {A GL(n, R AA T = 1, det A = 1} also has dimension (n 2 n/2. 9. Generalization: O(p, q, R orthogonal with respect to metric with signature p, q. e.g. O(3, 1 Lorentz group. (SO(p, q, R 2 Lie Algebra of a Lie Group Definition A Lie Algebra A is a vector space with an additional map A A A X 1, X 2 [X 1, X 2 ] such that [ax + by, Z] = a[x, Z] + b[y, Z] [X, Y ] = [Y, X] [X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y ]] = (Jacobi Identity 2
3 Example: VFs on a Manifold. Each Lie Group has an associated Lie Algebra which encodes many properties of the group (e.g. dimension, compactness, if G is simply connected every rep. of Lie alg. gives rep. of Lie group. Definition (4.3.1 A VF X on a Lie Group G is left-invariant if it is l g -related to itself for all g G, i.e. l g X = X l g X g = X gg g G g, g G Definition (4.3.2 A VF X on a Lie Group G is right-invariant if it is r g -related to itself for all g G, i.e. r g X = X r g X g = X g g g G g, g G The set of all left-invariant VFs is called L(G and is a VS. Fact (eq If VFs X 1 and X 2 on manifold M are h-related to VFs Y 1 and Y 2 on N (i.e. h X 1 = Y 1 then [X 1, X 2 ] is h-related to [Y 1, Y 2 ]. if X 1 and X 2 are left-invariant then l g [X 1, X 2 ] = [l g X 1, l g X 2 ] = [X 1, X 2 ] is also left-invariant. Therefore L(G is sub Lie algebra of the lie algebra of all VFs on G. Question: Are there any left-invariant VFs? Theorem (4.1 There exists an isomorphism i : T e G L(G, A L A. i given by L A g = l g A g G. This is left invariant because It is an isomorphism because: l g L a g = l g l g A = l g g A = L A g g. If L A = L B then L A e = L B e A = B and i is therefore injective. If L is left-invariant then L g = l g L e which is equal to L Le g. Therefore i is surjective. This means that dim L(G = dim T e G = dim G. Theorem (4.2 f : G H smooth homomorphism between Lie-Groups then f : L(G L(H is homomorphism between Lie Algebras. 3
4 Proof omitted. G f H?? L(G f L(H If {E 1, E 2,... E n } is basis of L(G then commutator must be linear combination of these: n [E α, E β ] = C γ αβ E γ C γ αβ γ=1 are called the structure constants of the Lie algebra. 2.1 Exponential Map Definition An integral curve of a VF X is a map σ : R G such that ( d σ = X σ(t. This means when applied to a coordinate function x i : G R ( ( d d σ (x i = (x i σ = d t t xi (σ(t = X σ(t (x i = Xσ(t i t t Definition (4.4 1 We call exp A : t exp ta the unique integral curve of the left invariant VF L A satisfying A = exp A ( d ( exp A = e. (A T e G This is defined for all t because every left-invariant VF is complete. (Not enough time for proof, idea is to extend curve by using group multiplication. Definition (4.4 2 The exponential map exp : T e G G is defined by exp A := exp ta t=1. It is a local diffeomorphism around e (in a neighbourhood around e it is bijective and it and its inverse are smooth. exp ta is a one parameter subgroup of G, i.e. it fulfils exp((t 1 + t 2 A = (exp t 1 A(exp t 2 A. In fact every one-parameter subgroup is of this form. Theorem (4.4 ( If χ : R G is one parameter subgroup then χ(t = exp ta with A := χ d. 4
5 Proof If χ : R G is a one-parameter subgroup then χ(t 1 + t 2 = χ(t 1 χ(t 2 ( χ( = e. This means χ l s = l χ(s χ s R (l s is add. with s in R. Therefore χ ( d s = χ l s ( d = l χ(s χ ( d = l χ(s (A = L A χ(s meaning t χ(t is integral curve for L A. But these are unique and therefore χ(t = exp ta. (unique because VF (tangent vector and one starting point given. Theorem (corollary If f : G H homomorphism between Lie groups G and H then f G H exp G exp H ( L(G f L(H commutes, i.e. exp H (f A = f(exp G A A T e G. Proof Def. χ : R H by χ(t := f(exp G ta. Then χ(t 1 + t 2 = f(exp G (t 1 + t 2 A = f(exp G t 1 A exp G t 2 A = f(exp G t 1 Af(exp G t 2 A = χ(t 1 χ(t 2 meaning χ is a one-parameter subgroup of H. This implies (by theorem 4.4 that it is given by ( d χ(t = exp H tb with B := χ T e H. ( Applying B to a function k C (H gives ( d B(k = (k χ = d k f exp G ta = L A e (k f t= Last step because exp G ta is integral curve of L A (see def. of integral curve. L A e = A so B(k = A(k f = (f A(k or B = f (A. Inserting this into equation gives f(exp G ta = χ(t = exp H tf (A which proves the theorem for t = 1. 5
6 Theorem (corollary If Ad g (g := gg g 1 g G then exp(ad g B = g exp(bg 1 ( Proof Ad g (e = e so Ad g maps T e G to T e G.For each g G, Ad g is a homomorphism of G, therefore applying the above theorem gives exp Ad g B = Ad g (exp B = g exp(bg 1 The map g Ad g gives a representation of the Lie-group onto the Lie algebra called the adjoint representation. 2.2 The Lie Algebra of GL(n, R Consider GL(n, R +. It is a subset of M(n, R and a natural system of coordinates are the matrix elements given by x i j : GL(n, R + R; x i j(g := g i j. Therefore the tangent space at every point is M(n, R. We want to find the explicit form of the lie algebra. The coordinate representation of the left invariant vector fields (i.e. the lie algebra is ( L A g = L A g (x i j. x i j The components of the vector field can be written as ( d L A g (x i j = (l g A(x i j = (l g (exp ta = ((g exp ta ( d (x i j = d g (x i j ( x i j (g exp ta. t= For matrices we can consider the curve t e ta, which is a one-parameter subgroup of GL(n, R + (e t 1A e t 2A = e (t 1+t 2 A and whose derivative at t = is A. This means e ta = exp ta t R A T e G = M(n, R. Inserting this into the expression for the components of the vector field L A gives L A g (x i j = d xi j(ge ta = d t= gi k(e ta k j t= = g i d k (eta k j = g i ka k j = (ga i j t= 6
7 So the left-invariant VF L A g has the local coordinate representation L A g = (ga i j ( x i j g. To understand the Lie algebra we also need the coordinate representation of the Lie bracket. Calculating the Lie bracket of the VFs L A and L B gives [L A, L B ] g = (ga i j( j i g(gb i j j i (gb i j( j i g(ga i j j i = g i ka k j( j i gi l g B l j ( i j g + g i ka k jg i lb l j ( j i j g This means that g i kb k j ( j i gi l g }{{} δi i δ j l A l j j ( g g i kb k j g i la l j ( j i i i j i g = g i ka k jb j j j i g i kb k j A j j j i = g i k[a, B] k j j i [L A, L B ] = L [A,B] i.e. the matrix commutator gives the Lie bracket. 2.3 Left-Invariant Forms = (g[a, B] i j j i. Analogous to left/right-invariant VFs, define left/right-invariant n-forms. Definition (4.5 An n-form ω is left-invariant if l gω = ω g G l g(ω g = ω g 1 g g, g G. Because pullbacks commute with the exterior derivative d this means l g(dω = d(l gω = dω, i.e. if ω is left-invariant then dω is too. The set of all left-invariant one-forms is denoted by L (G. We know the structure constants for left-invariant VFs: Define a dual basis ω 1, ω 2,..., ω n for L (G by [E α, E β ] = C γ αβ E γ. (4.3.5 w α, E β := δ α β. 7
8 The analogue of for one-forms is the Cartan-Maurer equation dω α Cα βγ ω β ω γ =. This contains the exterior derivative because while the lie bracket of two VFs gives another VF, the wedge product of two one-forms gives a two-form. Definition (4.6 The Cartan-Maurer form Ξ is the L(G valued one-form (Ξ : T G L(G on G such that Or equivalently Ξ, v g = v v T g G g G. Ξ, v g := l g (l g 1 v v T g G g, g G. The Cartan-Maurer form is left-invariant. Applying it to a left-invariant VF L A gives Ξ, L A g g = L A g. 3 Infinitesimal Transformations Definition (4.8 A right-action of a Lie-group G on a manifold M is a homomorphism δ : G Diff(M; g δ g i.e. δ e (p = p δ g (δ g (p = δ g g(p. such that the map G M (g, p δ g (p M is smooth. Often δ g (p is written as pg. The Homomorphism condition is then (pg 1 g 2 = p(g 1 g 2. Given such an action, every one-parameter subgroup of G gives a manifold-filling family of curves on M. These do not cross because mσ(t = m σ(t mσ(tσ( t = m σ(t σ( t mσ(t t = m. }{{} e No self intersection because mσ(t = mσ(t mσ(t + t = mσ(t + t. By taking the tangent vector to these curves this defines the induced vector field. Definition (4.1 If a Lie-group G has a right action on a manifold M then the VF X A on M induced by t exp ta is defined as with f C (M. X A p (f := d f(p exp ta t= 8
9 This means φ A t (p := p exp ta is a flow of X A. Define M p : G M p M M p (g := pg. Using this (M p L A g (f =L A g (f M p = (l g A(f M p = A(f M p l g = A(f M pg = d f(m pg(exp ta = d t= t= f(pg exp ta = Xpg(f A. Therefore M p L A g = X A pg and M p A = X A p (alternate definition of induced VF. Theorem (4.8 Lie-group G has right action on manifolds M, M with induced VFs X A, X A and f : M M is equivariant ( f(pg = f(pg p M, g G then f X A p = X A f(p Proof (f M p (g = f(pg = f(pg = M f(p(g f X A p = f M p A = (f M p A = M f(p A = X A f(p Special case: M = G with action δ g = r g Then M g (g = gg = l g (g X A g = M g A = l g A = L A g. So the left-invariant VFs are induced by right translation. From definition of induced VF: L A g (f = d f(g exp ta t=. This way of looking at the VFs L A leads to Theorem (4.9 For A, B T e G [L A, L B ] e = d Ad exp ta B Proof In general (eq if φ X t is a flow of X on M and Y some other VF then [X, Y ] = d φx t Y 1 ( = lim Y φ X t t= t t Y t= 9
10 Here φ A t = r exp ta and therefore [L A, L B 1 ( ] e = lim L B t t e r exp ta L B 1 exp ta = lim t t (B r exp ta l exp ta B 1 = lim t t (B Ad 1 exp ta B = lim t t (Ad exp ta B B = d Ad exp ta B t= Theorem (4.11 If a Lie-group G has a right action on a manifold M then A X A is a Lie-algebra homomorphism from L(G into Vfld(M i.e. [X A, X B ] = X [AB] := X [LA,L B ] e A, B T e G. This means a representation of the Lie-group gives a representation of the Lie-algebra. (This also requires that A X A is linear which is clear because X A p = M p A Proof First show X Adg A = δ g 1 X A : X Adg A p = M p Ad g A = (M p Ad g A (M p Ad g (g = p(gg g 1 = (δ g 1 M pg (g X Adg A p = (δ g 1 M pg A = δ g 1 X A pg Now use eq again with the flow δ exp ta for X A [X A, X B 1 ( ] = lim X B δ exp ta X B 1 ( = lim X B X Ad exp ta B t t t t 1 = lim t t XB Ad exp ta B 1 = lim t t XAd exp ta B B = X lim t (Ad exp ta B B/t = X [LA,L B ] e. The opposite direction (representation of algebra representation of group is possible if G is simply connected and M is compact ( Palais theorem. 1
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