Lecture 33: Lie Groups

Lecture 33: Lie Groups 33 33.1 Lie Groups Review Last time, we discussed one-parameter groups G ≤ GLn (R). We started thinking about tangent vectors to the group, based at some identity. Defnition 33.1 The collection of all tangent vectors at I ⊂ G, called the tangent space, can be characterized in multiple ways. We call this Lie(G), pronounced "lee." 1. The frst defnition is the most familiar one. The tangent vectors are the matrices such that the associated one-parameter subgroup lies in G. We found that there is a bijection between matrices and one-parameter groups lying in G. Lie(G) = {A ∈ Matn×n (R) : etA ∈ G, t ∈ R} 2. More generally, we consider any path inside the group through the identity, not just a one-parameter group, and take A, the velocity at the identity, to be a tangent vector.108 Lie(G) = {A ∈ Matn×n (R) : ∃f : (−ε, ε) −→ G, f (0) = I, f ′ (0) = A} 3. The third approach is slightly stranger. It is less general, and requires G to be defned by polynomial constraints.109 In this case, we take this strange construction R[ε] = R ⊕ Rε = {a + bε : a, b ∈ R}, where we defne the multiplication to be such that ε2 = 0.110 108 For defnition 1, we know all the 1-parameter subgroups, but for defnition 2, there are lots of other possible paths. So for defnition 1, there is a bijection between the matrices A and the one-parameter groups, while for defnition 2, there are lots of diferent paths with the same tangent vector as the velocity. Defnition 2 does not use the fact that G is a group. 109 For example, setting the determinant to be 1 is some complicated polynomial constraint. 110 This is similar to how we could defne the complex numbers, where we set i2 = −1. Lecture 33: Lie Groups If f is a polynomial, then we can set, formally, the derivative to be f ′ (x) = f (x + ε) − f (x). This is a way of thinking about the derivative without limits, for polnyomials. Then, we take Lie(G) = {A ∈ Matn×n : I + εA satisfes the polynomial constraints defning G}. 33.2 Lie Groups These defnitions are quite abstract, so let’s see them in action for On . Example 33.2 Let G = On , the set of matrices such that AT A = I. 1. The Lie group, as we have seen in the previous lecture, is Lie(On ) = {A : AT = −A}. These are the skew-symmetric matrices. 2. Consider a path passing through the identity at zero: f : (−ε, ε) −→ On such that f (0) = I. By the defnition of an orthogonal matrix, f (t)T · f (t) = I. Taking the derivative, f ′ (t)T · f (t) = f (t)T · f ′ (t) = 0, and taking t = 0 gives AT I + IA = 0, and thus AT = −A. So the same condition holds. 3. The condition that AT A = I is a set of complicated polynomial conditions. From defnition 3), the Lie group consists of matrices A such that (I + εA)T (I + εA) = I, using the rule that ε2 = 0. Multiplying this out, I + εAT + εA + ε2 AT A, and taking ε2 = 0, which implies that I + εAT + εA = I, AT = −A after dividing both sides by ε. All three defnitions lead to the same Lie group, despite being very diferent. The third defnition is useful because it makes sense even without working over the real numbers, and works for any group defned by polynomial constraints!111 For example, the Lie group can be defned for orthogonal matrices over fnite felds. Here are some non-obvious facts about these characterizations of the tangent space at the identity. 111 The intuition for this third defnition is that it is essentially using the Taylor expansion of the path through the identity, and ignoring third order and higher terms. Lecture 33: Lie Groups Proposition 33.3 For Lie(G) : • All three defnitions are actually equivalent. • For a group G, Lie(G) is actually a vector subspace of Matn×n (R). It is surprising that Lie(G) is a vector space, since the matrix exponential does not generally behave well with respect to addition if A and B do not commute.112 33.3 Manifolds In order to understand Lie groups, we have to think about the notion of a manifold. For this section, the discussion will be less rigorous and precise, and it is okay not to understand all the defnitions; we are just providing the favor of concepts that will show up in later classes. Defnition 33.4 For M a subset of Rn , M is a (diferentiable) manifold of dimension d if for each x ∈ M, there exists an open set containing x V ⊆ M , an open ball U ⊂ Rda , and a continuous (diferentiable) bijection f : U −→ V. a An open ball is a subset of Rd of the form U = {x : |x| < δ} for some δ. Globally, a d-dimensional does not look like Rd , but locally, it does. The circle is an example of a 1-dimensional manifold; at each point on the circle, there is really only one direction to move in. Example 33.5 (Circle) Consider some interval on the real line. Then, it is possible to write down some function bijectively mapping that interval to some other interval on the circle. This can happen around any point on the circle, so it is a manifold. 112 Using the frst defnition, it is not clear that etA + etB can be written as etC . Lecture 33: Lie Groups Loosely, for defnition 2, we have tangent vectors at 0 in U ⊂ Rd corresponding to tangent vectors at x in M, which in a way brings the vector space structure from Rd to M. A non-example would be the union of the x-axis and the y-axis, since at the origin, there is an intersection that does not look like an interval. There are two directions to move in, instead of one direction. All our examples of G ≤ GLn (R) are manifolds, not just groups. This requires some argument, but it is true. 33.4 Lie Bracket For every group G, there is a corresponding vector space structure Lie(G) ⊂ Matn×n (R) = Lie(GLn (R)). In fact, Lie(G) carries some extra structure. The multiplication structure A, B ⇝ AB ∈ Matn×n (R) does not preserve Lie(G); if A, B ∈ Lie(G), it does not mean that AB ∈ Lie(G). For example, Lie(On ) = {A : AT = −A}, so for A, B ∈ Lie(G), (AB)T = B T AT = BA, which is not usually equal to −AB. However, a very similar structure does preserve the Lie group. Defnition 33.6 (Lie Bracket) Let the Lie bracket of A, B ∈ Matn×n (R) be [A, B] := AB − BA ∈ Matn×n (R). Theorem 33.7 For any G ≤ GLn , the Lie group G ⊆ Matn×n (R) is preserved by the Lie bracket: A, B ∈ Lie(G) → [A, B] ∈ Lie(G) The Lie group is not just a vector space — it is actually a vector space with some weirdo multiplication on it! The Lie bracket can be seen in action for some of the Lie groups we have seen already. Example 33.8 For G = On , Lie(On ) = {A : AT = −A}. Then for A, B ∈ Lie(On ), [A, B] = AB − BA, and [A, B]T = B T AT − AT B T = BA − AB = −[A, B]. So [A, B] ∈ Lie(On ). Example 33.9 For SLn (R), Lie(SLn ()) = {A : trace(A) = 0}. For matrices A, B, trace(AB) = trace(BA), and so trace([A, B]) = 0. The commutator in the group corresponds to the Lie bracket. Lecture 33: Lie Groups Proof. For A, B ∈ Lie(G), then consider etA ∈ G and esB ∈ G. Then we know that etA esB e−tA e−sB ∈ G, and Taylor expanding gives (I + tA + · · · )(I + sB + · · · )(I − tA + · · · )(I − sB + · · · ) ∈ G, which, to the frst order, gives I + st[A, B] + · · · ∈ G. Then, taking the derivative at 0 gives [A, B] ∈ Lie(G). The Lie bracket comes from the fact that the group is closed under multiplication and taking the derivative at 0 for that. As a corollary, if G is abelian113 , then the Lie bracket is identically 0 on all of G. In some way, for an arbitrary group G, the Lie bracket "measures" the failure of the group to be abelian. We started out with groups that we cared about (matrix groups), looked at the set of tangent vectors, which is the same as the set of one-parameter groups, and looked at the vector space structure which also has a funky Lie bracket, and now we will look at properties of this bracket. Here are some nice properties of the Lie bracket. • Antisymmetry. [A, B] = −[B, A] • The Jacobi identity. We have [[A, B], C] + [[B, C], A] + [[C, A], B] = 0. It is true simply when expanding. This also comes from a property of the group; we won’t do it but you can get it by staring at a more complicated version of how we derived the Lie bracket. Defnition 33.10 A Lie algebra (over R) is a vector space V with a Lie bracket [·, ·] : V ×V −→ V satisfying [A, B] = −[B, A] and the Jacobi identity. On the homework, we see that R2 with the cross product is a Lie algebra, and it is Lie(SO3 ) = Lie(SU2 ). If G is a group that is also a manifold, it is called a Lie group, and if G is a Lie group, it can be replaced with Lie(G), a Lie algebra, which is simply a vector space with a weird multiplication on it, which is a lot easier to study. However, the Lie algebra carries a lot of information about G. Theorem 33.11 Given a Lie algebra (fnite dimensional over R) V , there exists a unique Lie group G such that Lie(G) = V. Lie theory ends up being a very powerful tool in the study of understanding groups. 113 For example, diagonal matrices