Lecture 13 - Lie groups and their Lie algebras (Schuller's Geometric Anatomy of Theoretical Physics)

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Lie groups and their Lie algebras

Lie theory is a topic of the utmost importance in both physics and differential geometry. 1.1

Lie groups

Definition. A Lie group is a group (G, •), where G is a smooth manifold and the maps µ: G × G → G (g1 , g2 ) 7→ g1 • g2 and i: G → G g 7→ g −1 are both smooth. Note that G × G inherits a smooth atlas from the smooth atlas of G. Definition. The dimension of a Lie group (G, •) is the dimension of G as a manifold. Example 1.1. a) Consider (Rn , +), where Rn is understood as a smooth n-dimensional manifold. This is a commutative (or abelian) Lie group (since • is commutative), often called the n-dimensional translation group. b) Let S 1 := {z ∈ C | |z| = 1} and let · be the usual multiplication of complex numbers. Then (S 1 , ·) is a commutative Lie group usually denoted U(1). ∼

c) Let GL(n, R) = {φ : Rn − → Rn | det φ 6= 0}. This set can be endowed with the structure of a smooth n2 -dimensional manifold, by noting that there is a bijection ∼ between linear maps φ : Rn − → Rn and R2n . The condition det φ 6= 0 is a so-called open condition, meaning that GL(n, R) can be identified with an open subset of R2n , from which it then inherits a smooth structure. Then, (GL(n, R), ◦) is a Lie group called the general linear group. d) Let V be an n-dimensional R-vector space equipped with a pseudo inner product, i.e. an bilinear map (−, −) : V × V → R satisfying i) symmetry: ∀ v, w ∈ V : (v, w) = (w, v); ii) non-degeneracy: (∀ w ∈ V : (v, w) = 0) ⇒ v = 0. Ordinary inner products satisfy a stronger condition than non-degeneracy, called positive definiteness, which is ∀ v ∈ V : (v, v) ≥ 0 and (v, v) = 0 ⇒ v = 0. Given a symmetric bilinear map (−, −) on V , there is always a basis {ea } of V such that (ea , ea ) = ±1 and zero otherwise. If we get p-many 1s and q-many −1s (with p + q = n, of course), then the pair (p, q) is called the signature of the map. Positive definiteness is the requirement that the signature be (n, 0), although in relativity we

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require the signature to be (n − 1, 1). A theorem states that there are (up to isomorphism) only as many pseudo inner products on V as there are different signatures. We can define the set ∼

O(p, q) := {φ : V − → V | ∀ v, w ∈ V : (φ(v), φ(w)) = (v, w)}. The pair (O(p, q), ◦) is a Lie group called the orthogonal group with respect to the pseudo inner product (−, −). This is, in fact, a Lie subgroup of GL(p + q, R). Some notable examples are O(3, 1), which is known as the Lorentz group in relativity, and O(3, 0), which is the 3-dimensional rotation group. Definition. Let (G, •) and (H, ◦) be Lie groups. A map φ : G → H is Lie group homomorphism if it is a group homomorphism and a smooth map. A Lie group isomorphism is a group homomorphism which is also a diffeomorphism. 1.2

The left translation map

To every element of a Lie group there is associated a special map. Note that everything we will do here can be done equivalently by using right translation maps. Definition. Let (G, •) be a Lie group and let g ∈ G. The map `g : G → G h 7→ `g (h) := g • h ≡ gh is called the left translation by g. If there is no danger of confusion, we usually suppress the • notation. Proposition 1.2. Let G be a Lie group. For any g ∈ G, the left translation map `g : G → G is a diffeomorphism. Proof. Let h, h0 ∈ G. Then, we have `g (h) = `g (h0 ) ⇔ gh = gh0 ⇔ h = h0 . Moreover, for any h ∈ G, we have g −1 h ∈ G and `g (g −1 h) = gg −1 h = h. Therefore, `g is a bijection on G. Note that `g = µ(g, −) and since µ : G × G → G is smooth by definition, so is `g . The inverse map is (`g )−1 = `g−1 , since `g−1 ◦ `g = `g ◦ `g−1 = idG . Then, for the same reason as above with g replaced by g −1 , the inverse map (`g )−1 is also smooth. Hence, the map `g is indeed a diffeomorphism.

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Note that, in general, `g is not an isomorphism of groups, i.e. `g (hh0 ) 6= `g (h) `g (h0 ) in general. However, as the final part of the previous proof suggests, we do have `g ◦ `h = `gh for all g, h ∈ G. Since `g : G → G is a diffeomorphism, we have a well-defined push-forward map (Lg )∗ : Γ(T G) → Γ(T G) X 7→ (Lg )∗ (X) where (Lg )∗ (X) : G → T G h 7→ (Lg )∗ (X)(h) := (`g )∗ (X(g −1 h)). We can draw the diagram TG

(`g )∗

TG (Lg )∗ (X)

X

G

`g

G

Note that this is exactly the same as our previous Φ∗ (σ) := φ∗ ◦ σ ◦ φ−1 . By introducing the notation X|h := X(h), so that X|h ∈ Th G, we can write (Lg )∗ (X)|h := (`g )∗ (X|g−1 h ). Alternatively, recalling that the map `g is a diffeomorphism and relabelling the elements of G, we can write this as (Lg )∗ (X)|gh := (`g )∗ (X|h ). A further reformulation comes from considering the vector field X ∈ Γ(T G) as an R-linear ∼ map X : C ∞ (G) − → C ∞ (G). Then, for any f ∈ C ∞ (G) (Lg )∗ (X)(f ) := X(f ◦ `g ). Proposition 1.3. Let G be a Lie group. For any g, h ∈ G, we have (Lg )∗ ◦ (Lh )∗ = (Lgh )∗ .

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Proof. Let f ∈ C ∞ (G). Then, we have

(Lg )∗ ◦ (Lh )∗ (X)(f ) = (Lg )∗ (Lh )∗ (X) (f ) = (Lh )∗ (X)(f ◦ `g ) = X(f ◦ `g ◦ `h ) = X(f ◦ `gh ) =: (Lgh )∗ (X)(f ), as we wanted to show. The previous identity applies to the pointwise push-forward as well, i.e.
(`g1 )∗ ◦ (`g2 )∗ (X|h ) = (`g1 g2 )∗ (X|h ) for any g1 , g2 , h ∈ G and X|h ∈ Th G. 1.3

The Lie algebra of a Lie group

In Lie theory, we are typically not interested in general vector fields, but rather on special class of vector fields which are invariant under the induced push-forward of the left translation maps `g . Definition. Let G be a Lie group. A vector field X ∈ Γ(T G) is said to be left-invariant if ∀ g ∈ G : (Lg )∗ (X) = X. Equivalently, we can require this to hold pointwise ∀ g, h ∈ G : (`g )∗ (X|h ) = X|gh . By recalling the last reformulation of the push-forward, we have that X ∈ Γ(T G) is left-invariant if, and only if ∀ f ∈ C ∞ (G) : X(f ◦ `g ) = X(f ) ◦ `g . We denote the set of all left-invariant vector fields on G as L(G). Of course, L(G) ⊆ Γ(T G) but, in fact, more is true. One can check that L(G) is closed under + : L(G) × L(G) → L(G) · : C ∞ (G) × L(G) → L(G), only for the constant functions in C ∞ (G). Therefore, L(G) is not a C ∞ (G)-submodule of Γ(T G), but it is an R-vector subspace of Γ(T G). Recall that, up to now, we have refrained from thinking of Γ(T G) as an R-vector space since it is infinite-dimensional and, even worse, a basis is in general uncountable. A priori, this could be true for L(G) as well, but we will see that the situation is, in fact, much nicer as L(G) will turn out to be a finite-dimensional vector space over R.

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Theorem 1.4. Let G be a Lie group with identity element e ∈ G. Then L(G) ∼ =vec Te G. ∼

Proof. We will construct a linear isomorphism j : Te G − → L(G). Define j : Te G → Γ(T G) A 7→ j(A), where j(A) : G → T G g 7→ j(A)|g := (`g )∗ (A). i) First, we show that for any A ∈ Te G, j(A) is a smooth vector field on G. It suffices to check that for any f ∈ C ∞ (G), we have j(A)(f ) ∈ C ∞ (G). Indeed (j(A)(f ))(g) = j(A)|g (f ) := (`g )∗ (A)(f ) = A(f ◦ `g ) = (f ◦ `g ◦ γ)0 (0), where γ is a curve through e ∈ G whose tangent vector at e is A. The map ϕ: R × G → R (t, g) 7→ ϕ(t, g) := (f ◦ `g ◦ γ)(t) = f (gγ(t)) is a composition of smooth maps, hence it is smooth. Then (j(A)(f ))(g) = (∂1 ϕ)(0, g) depends smoothly on g and thus j(A)(f ) ∈ C ∞ (G). ii) Let g, h ∈ G. Then, for every A ∈ Te G, we have (`g )∗ (j(A)|h ) := (`g )∗ ((`h )∗ (A)) = (`gh )∗ (A) = j(A)|gh , so j(A) ∈ L(G). Hence, the map j is really j : Te G → L(G). iii) Let A, B ∈ Te G and λ ∈ R. Then, for any g ∈ G j(λA + B)|g = (`g )∗ (λA + B) = λ(`g )∗ (A) + (`g )∗ (B) = λj(A)|g + j(B)|g , ∼

since the push-forward is an R-linear map. Hence, we have j : Te G − → L(G).

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iv) Let A, B ∈ Te G. Then j(A) = j(B) ⇔ ∀ g ∈ G : j(A)|g = j(B)|g ⇒ j(A)|e = j(B)|e ⇔ (`e )∗ (A) = (`e )∗ (B) ⇔ A = B, since (`e )∗ = idT G . Hence, the map j is injective. v) Let X ∈ L(G). Define AX := X|e ∈ Te G. Then, we have j(AX )|g = (`g )∗ (AX ) = (`g )∗ (X|e ) = Xge = Xg , since X is left-invariant. Hence X = j(AX ) and thus j is surjective. ∼

Therefore, j : Te G − → L(G) is indeed a linear isomorphism. Corollary 1.5. The space L(G) is finite-dimensional and dim L(G) = dim G. We will soon see that the identification of L(G) and Te G goes beyond the level of linear isomorphism, as they are isomorphic as Lie algebras. Recall that a Lie algebra over an algebraic field K is a vector space over K equipped with a Lie bracket [−, −], i.e. a K-bilinear, antisymmetric map which satisfies the Jacobi identity. Given X, Y ∈ Γ(T M ), we defined their Lie bracket, or commutator, as [X, Y ](f ) := X(Y (f )) − Y (X(f )) for any f ∈ C ∞ (M ). You can check that indeed [X, Y ] ∈ Γ(T M ), and that the bracket is R-bilinear, antisymmetric and satisfies the Jacobi identity. Thus, (Γ(T M ), +, ·, [−, −]) is an infinite-dimensional Lie algebra over R. We suppress the + and · when they are clear from the context. In the case of a manifold that is also a Lie group, we have the following. Theorem 1.6. Let G be a Lie group. Then L(G) is a Lie subalgebra of Γ(T G). Proof. A Lie subalgebra of a Lie algebra is simply a vector subspace which is closed under the action of the Lie bracket. Therefore, we only need to check that ∀ X, Y ∈ L(G) : [X, Y ] ∈ L(G). Let X, Y ∈ L(G). For any g ∈ G and f ∈ C ∞ (G), we have [X, Y ](f ◦ `g ) := X(Y (f ◦ `g )) − Y (X(f ◦ `g )) = X(Y (f ) ◦ `g ) − Y (X(f ) ◦ `g ) = X(Y (f )) ◦ `g − Y (X(f )) ◦ `g
= X(Y (f )) − Y (X(f )) ◦ `g = [X, Y ](f ) ◦ `g . Hence, [X, Y ] is left-invariant.

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Definition. Let G be a Lie group. The associated Lie algebra of G is L(G). Note L(G) is a rather complicated object, since its elements are vector fields, hence we would like to work with Te G instead, whose elements are tangent vectors. Indeed, we can use the bracket on L(G) to define a bracket on Te G such that they be isomorphic as Lie algebras. First, let us define the isomorphism of Lie algebras. Definition. Let (L1 , [−, −]L1 ) and (L2 , [−, −]L2 ) be Lie algebras over the same field. A ∼ → L2 is a Lie algebra homomorphism if linear map φ : L1 − ∀ x, y ∈ L1 : φ([x, y]L1 ) = [φ(x), φ(y)]L2 . If φ is bijective, then it is a Lie algebra isomorphism and we write L1 ∼ =Lie alg L2 . By using the bracket [−, −]L(G) on L(G) we can define, for any A, B ∈ Te G
[A, B]Te G := j −1 [j(A), j(B)]L(G) , where j −1 (X) = X|e . Equipped with these brackets, we have L(G) ∼ =Lie alg Te G.

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