Lecture 30: The Special Unitary Group SU2

Lecture 30: The Special Unitary Group and One-Parameter Groups 30 30.1 The Special Unitary Group SU2 Review Last time, we started looking at subgroups of the group of invertible matrices. We saw that one thing these groups have in common, that fnite or discrete groups don’t, is that they have some sort of shape or geometry. In particular, we looked at the group SU2 := {A ∈ GL2 (C) | A∗ = A−1 , det A = 1}, the special unitary group. By playing around with the defnitions, we found that SU2 sits inside the quaternions H = {x0 I + x1 i + x2 j + x3 k}, where we defned  i i= 0   0 0 ,j= −i −1   1 0 ,k= 0 i  i . 0 In particular, SU2 is the subset with x20 + x21 + x22 + x23 = 1, corresponding to the 3-sphere S 3 in R4 . Note 30.1 We’ve seen that the 1-dimensional sphere and 3-dimensional spheres both have a group structure, so we can ask whether the same is true for other dimensions. It turns out the answer is no – there are no other n-dimensional spheres which can be made into groups. (This is a deeper fact.) Last class, we started taking geometric properties of the 3-sphere and seeing how they correspond to the group structure. In particular, we looked at the latitudes – the horizontal slices Latc = {x0 = c} ∩ S 3 for −1 ≤ c ≤ 1. We proved last class that these latitudes are precisely the conjugacy classes of SU2 . We call Lat0 the equator, denoted E. 30.2 Longitudes Another thing we can think about are the longitudes – the circles which go through the north and south pole. We can defne these more precisely: for each x ∈ E, the longitude containing x is Longx := Span(I, x) ∩ S 3 . Here Span(I, x) is a 2-dimensional plane, so we’re taking the unit circle of a 2-dimensional plane. Theorem 30.2 For each x ∈ E, Longx is a subgroup of SU2 . In fact, given θ ∈ R/2πZ, the map θ 7→ cos θI + sin θx is an isomorphism between R/2πZ and Longx . What this means is that the longitudes aren’t just circles as shapes, they’re also circles as groups (since we’ve seen that the unit circle as a group is isomorphic to R/2πZ. Proof. To see this is true, we’ll frst consider the special case x = i. Then we can check that given two points in Longi , we have (c + si)(c′ + s′ i) ∈ Span(I, i) Lecture 30: The Special Unitary Group and One-Parameter Groups as well, since i2 = −I. Meanwhile, since both elements are in SU2 , their product must be as well; so their product is in SU2 ∩ Span(I, i) = Longi . So this longitude is closed under multiplication, and is therefore a subgroup. We won’t check the isomorphism to R/2πZ here, but it’s possible to check this directly by multiplying out. We can then use this to solve the general case – for any x ∈ E, we know x is conjugate to i (since we saw that the equator is a conjugacy class). So then we can write x = P −1 iP , and then Longx = P −1 Longi P is conjugate to Longi . But when we conjugate a subgroup, we get another subgroup. So not only are the longitudes all circle subgroups, but they’re also all conjugate to each other. 30.3 More Group Theoretic Properties When we studied conjugacy classes, we also studied centralizers: Guiding Question What is the centralizer of i? Recall that this means the set of elements for which if we conjugate i by them, we get back i. We know that Longi is a subgroup of SU2 , and it’s abelian (since it’s isomorphic to the circle). Since i is in this longitude, this means it commutes with everything in this longitude. So Z(i) ⊃ Longi . In fact, this turns out to be an equality – we have Z(i) = Longi . This is true for any other point on the equator as well – its longitude is exactly its centralizer. Another thing we saw when studying conjugacy classes was that there’s a bijection between the conjugacy class C(g) and the cosets of the centralizer G/Z(g). In our case, this is still true, but now both sides are geometric objects. If we fx a point g on the equator, then C(g) = E is a 2-dimensional sphere. Meanwhile, G/Z(g) corresponds to taking cosets of a longitude. We’re taking a 3-sphere and covering it in cosets – so we have a map S 3 → S 2 , where the fbers are circles (the cosets of the longitude). This is really hard to picture, but the idea is that we start with the 3-sphere and a given longitude, and we’re taking its cosets (which correspond to circles not necessarily through the north and south pole) and covering the entire 3-sphere in these circles. When we collapse all these circles to a point, we get a copy of the 2-sphere. Note 30.3 This is really difcult to think about, but it’s a construction in topology relating spheres of dimensions 1, 2, and 3, and it can also be thought of in this group theory setting. What we would like to illustrate is that group theoretic facts about this group also become interesting geometric facts; it doesn’t really matter if you don’t understand all of them. 30.4 Conjugation and the Orthogonal Group There’s another thing we can look at: we know the equator is a conjugacy class, so SU2 acts on E transitively (with the action given by conjugation). In fact, SU2 acts on the space {x0 = 0} ⊂ H (which is the 3-dimensional vector space containing the equator), and it preserves the equator E inside this space. Conjugation by an element of SU2 is a linear map, so it defnes a group homomorphism ρ : SU2 → GL3 (R), where ρ(g) is the matrix such that ρ(g)⃗v = gvg −1 . But ρ(g) preserves E, so since it preserves vectors of length 1, this means it must preserve length in general. So ρ(g) is actually an isometry – which means this map is actually ρ : SU2 → O3 . In fact, we can say even more. We’ve seen that orthogonal matrices in 3 dimensions are either refections or rotations – and you can tell which by looking at the determinant (which is always ±1). But SU2 is connected (we can get from any point to any other point by following some path), so det(ρ(g)) can’t jump between ±1 (since ρ is continuous). So then det(ρ(g)) is constant as g varies. We know that det(ρ(I)) = 1, so then det(ρ(g)) is always 1. So in fact, this is a homomorphism ρ : SU2 → SO3 . Lecture 30: The Special Unitary Group and One-Parameter Groups Note 30.4 We could write down this homomorphism in terms of the matrix entries – we start with a 2 × 2 complex matrix and create a 3 × 3 real one, and we could explicitly write down the homomorphism. But it’s more interesting to think about it geometrically, by considering the action of SU2 on one of its conjugacy classes. Note 30.5 You can go further with this – given a point on the 3-sphere, we can ask how to fgure out what angle and axis of rotation it corresponds to. This is written up in the notes, but we won’t discuss it here. But you can go really far by playing around with the group-theoretic constructions we’ve seen earlier and trying to picture what they mean. Student Question. Did we show that ρ was continuous? Answer. No, we did not. In order to check that it’s continuous, you can write down the map in terms of the entries. But this shouldn’t be surprising – we have ρ(g)v = gvg −1 , and we can write down a explicit formula for g −1 in terms of g. So the matrix ρ(g) is something we can write down explicitly in terms of coordinates. (In fact, you can also use this explicit formula to show it’s in SO3 , but it’s nicer to just show that it’s continuous and then deduce it’s in SO3 by thinking about it geometrically.) Student Question. Was the action SU2 defned on E just left multiplication? Answer. No, it’s conjugation. The idea is that E is one of the conjugacy classes of the group, and all the conjugacy classes are orbits with respect to the conjugation action. (This is why the action is transitive as well.) 30.5 One-Parameter Groups Now we’ll return to looking at linear groups more generally – subgroups G of GLn (R) or GLn (C) which satisfy some condition (for example, preserving volume or a bilinear form). Defnition 30.6 A one-parameter group (in GLn (R) or GLn (C)) is a diferentiable homomorphism from R → GLn (R) or R → GLn (C). In otherwords, it’s a function φ : R → GLn (C) with t 7→ φ(t). It should be a group homomorphism, so 2 φ(s + t) = φ(s) + φ(t), and it should be diferentiable (where we think of GLn (C) as sitting inside R2n – then each entry of the function should be a diferentiable function on R). One way to think of this defnition is as an analog of when we looked at maps Z → G. The integers are in some sense the simplest group we can write down – it has just one generator and no relations – and we can look at maps Z → G, to help us study G. The idea here is that (R, +) is basically the simplest one-dimensional group. (We haven’t defned dimension, but you can think of dimension as how many parameters we have. There are other one-dimensional groups, like a circle, but the real numbers are simpler because we don’t have relations like 2π = 0 here.) We’ve already seen a few examples of one-parameter groups: Example 30.7 In SU2 , the map θ 7→ cos θI + sin θx (for any x ∈ E) is a one-parameter group. These one-parameter groups are the longitudes. We’ve seen that every point lies in some longitude, and therefore some one-parameter group; in general, that isn’t always true. Let’s see another example, when n = 1. Example 30.8 When n = 1, the map φ : R → C× with φ(t) = eαt (for any α ∈ C) is a one-parameter group. Lecture 30: The Special Unitary Group and One-Parameter Groups Here one-dimensional matrices are just numbers. This construction works because we have φ(s + t) = eαs+αt = eαs eαt = φ(s)φ(t). Note 30.9 This isn’t an analysis class, so we won’t check the diferentiability of these maps. In this example, it can be done by writing everything down in terms of sines and cosines. Guiding Question Is there a version of this construction for n > 1? The answer is yes – if we have A ∈ Matn×n (C), we can try to defne eA . Taking a number to the power of a matrix doesn’t make any sense, but the exponential function also has a description using power series: we have ex = 1 + x + x2 x3 + + ··· . 2! 3! This is a very nice power series – it converges to ex everywhere. In fact, you can even take this as the defnition of ex , and you can take the derivative term-by-term. So we can use this to defne eA as well: Defnition 30.10 The exponential of a matrix A ∈ Matn×n (C) is eA = I + A + A3 A2 + + · · · ∈ Matn×n (C). 2! 3! This also converges uniformly as A varies in a bounded region, meaning that for every entry of the matrix, if we take the corresponding entries of each term, then we get a convergent series. As we vary A in a neighborhood, this convergence of that series is uniform. So this gives us a well-defned n×n matrix eA . (To be more precise, we can actually put a metric on the space of matrices, and use this to be careful about the notion of convergence.) This exponential has several nice properties, similarly to the normal exponential. • The exponential interacts well with conjugation – we have P −1 eA P = P −1 IP + P −1 AP + P −1 A2 /2P + · · · = I + P −1 AP + = eP −1 AP (P −1 AP )2 + ··· 2 . (To be more careful, the LHS and RHS are both defned by limits – where we take the power series and truncate it. The equality is true on the level of these truncations, so the limits are equal as well.) • If v is an eigenvector of A with eigenvalue λ, then v is also an eigenvector of eA with eigenvalue eλ . • We have esA etA = e(s+t)A . To prove this, we can expand the RHS out using the Binomial Theorem as X (s + t)n An X sk tℓ (k + ℓ)! k ℓ = · A A. n! (k + ℓ)! k!ℓ! n≥0 k,ℓ≥0 Then using uniform convergence, we can factor out the infnite sum as    X t ℓ Aℓ X sk Ak    = esA etA k! ℓ! k≥0 ℓ≥0 (the fact that we can rearrange in this way is the result of the strong convergence properties). In particular, eA e−A = I, so we actually have eA ∈ GLn (C). (We’re working in the complex case, but this works equally well in the real case.) Lecture 30: The Special Unitary Group and One-Parameter Groups The last result means that for any A ∈ Matn×n (C), φ(t) = etA is a one-parameter group in GLn (C). We can ask two questions about these one-parameter groups: Guiding Question Is every one-parameter group of this form? The answer will be yes, and we will see why in future lectures! Guiding Question Given a subgroup G ≤ GLn , what are the one-parameter subgroups living inside of G? We will discuss this question in future lectures as well.