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Chapter 1 Compact Groups

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Chapter 1 Compact Groups Chapter 1 Compact Groups Most infinite groups, in practice, come dressed in a natural topology, with re- spect to which the group operations are continuous. All the familiar groups— in particular, all matrix groups—are locally compact; and this marks the natural boundary of representation theory. A topological group G is a topological space with a group structure defined on it, such that the group operations 1 (x; y) xy; x x− 7! 7! of multiplication and inversion are both continuous. Examples: 1. The real numbers R form a topological group under addition, with the usual topology defined by the metric d(x; y) = x y : j − j 2. The non-zero reals R× = R 0 form a topological group under multipli- cation, under the same metric.n f g 3. The strictly-positive reals R+ = x R : x > 0 form a closed subgroup f 2 g of R×, and so constitute a topological group in their own right. Remarks: (a) Note that in the theory of topological groups, we are only concerned with closed subgroups. When we speak of a subgroup of a topological group, it is understood that we mean a closed subgroup, unless the contrary is explicitly stated. 424–II 1–1 424–II 1–2 (b) Note too that if a subgroup H G is open then it is also closed. For the cosets gH are all open; and⊂ so H, as the complement of the union of all other cosets, is closed. + So for example, the subgroup R R× is both open and closed. ⊂ Recall that a space X is said to be compact if it is hausdorff and every open covering X = U [ i ı I 2 has a finite subcovering: X = Ui Ui : 1 [···[ r (The space X is hausdorff if given any 2 points x; y X there exist open sets U; V X such that 2 ⊂ x U; y VU V = : 2 2 \ ; All the spaces we meet will be hausdorff; and we will use the term ‘space’ or ‘topological space’ henceforth to mean hausdorff space.) In fact all the groups and other spaces we meet will be subspaces of euclidean space En. In such a case it is usually easy to determine compactness, since a subspace X En is compact if and only if ⊂ 1. X is closed; and 2. X is bounded Examples: 1. The orthogonal group O(n) = T Mat(n; R): T 0T = I : f 2 g Here Mat(n; R) denotes the space of all n n real matrices; and T 0 denotes the transpose of T : × Tij0 = Tji: We can identify Mat(n; R) with the Euclidean space En2 , by regarding the 2 n entries tij as the coordinates of T . With this understanding, O(n) is a closed subspace of En2 , since it is the set of ‘points’ satisfying the simultaneous polynomial equations making up the matrix identity T 0T = I. It is bounded because each entry tij 1: j j ≤ 424–II 1–3 In fact, for each i, 2 2 2 t + t + + t = (T 0T )ii = 1: 1i 2i ··· ni Thus the orthogonal group O(n) is compact. 2. The special orthogonal group SO(n) = T O(n) : det T = 1 f 2 g is a closed subgroup of the compact group O(n), and so is itself compact. Note that T O(n) = det T = 1; 2 ) ± since 2 T 0T = I = det T 0 det T = 1 = (det T ) = 1; ) ) since det T 0 = det T . Thus O(n) splits into 2 parts: SO(n) where det T = 1; and a second part where det T = 1. If det T = 1 then it is easy to see that this second part is just the coset−T SO(n) of SO−(n) in O(n). We shall find that the groups SO(n) play a more important part in represen- tation theory than the full orthogonal groups O(n). 3. The unitary group U(n) = T Mat(n; C): T ∗T = I : f 2 g Here Mat(n; C) denotes the space of n n complex matrices; and T ∗ denotes the conjugate transpose of T : × Tij∗ = Tji: We can identify Mat(n; C) with the Euclidean space E2n2 , by regarding 2 the real and imaginary parts of the n entries tij as the coordinates of T . With this understanding, U(n) is a closed subspace of E2n2 . It is bounded because each entry has absolute value tij 1: j j ≤ In fact, for each i, 2 2 2 t1i + t2i + + tni = (T ∗T )ii = 1: j j j j ··· j j Thus the unitary group U(n) is compact. 424–II 1–4 When n = 1, U(1) = x C : x = 1 : f 2 j j g Thus 1 T1 R Z U(1) = S ∼= = = : Note that this group (which we can denote equally well by U(1) or T1) is abelian (or commutative). 4. The special unitary group SU(n) = T U(n) : det T = 1 f 2 g is a closed subgroup of the compact group U(n), and so is itself compact. Note that T U(n) = det T = 1: 2 ) j j since 2 T ∗T = I = det T ∗ det T = 1 = det T = 1; ) ) j j since det T ∗ = det T . The map U(1) SU(n) U(n):(λ, T ) λT × ! 7! is a surjective homomorphism. It is not bijective, since λI SU(n) λn = 1: 2 () Thus the homomorphism has kernel Cn = ! ; h i where ! = e2π=n. It follows that U(n) = (U(1) SU(n)) =Cn: × We shall find that the groups SU(n) play a more important part in repre- sentation theory than the full unitary groups U(n). 5. The symplectic group Sp(n) = T Mat(n; H): T ∗T = I : f 2 g Here Mat(n; H) denotes the space of n n matrices with quaternion en- × tries; and T ∗ denotes the conjugate transpose of T : ¯ Tij∗ = Tji: 424–II 1–5 (Recall that the conjugate of the quaternion q = t + xi + yj + zk is the quaternion q¯ = t xi yj zk: − − − Note that conjugacy is an anti-automorphism, ie q1q2 = q2q1: It follows from this that (AB)∗ = B∗A∗ for any 2 matrices A; B whose product is defined. This in turn justifies our implicit assertion that Sp(n) is a group: S; T Sp(n) = (ST )∗(ST ) = T ∗S∗ST = T ∗T = I = ST Sp(n): 2 ) ) 2 Note too that while multiplication of quaternions is not in general commu- tative, q and q¯ do commute: qq¯ = qq¯ = t2 + x2 + y2 + z2 = q 2; j j defining the norm, or absolute value, q of a quaternion q.) j j We can identify Mat(n; H) with the Euclidean space E4n2 , by regarding 2 the coefficients of 1; i; j; k in the n entries tij as the coordinates of T . With this understanding, Sp(n) is a closed subspace of E4n2 . It is bounded because each entry has absolute value tij 1: j j ≤ In fact, for each i, 2 2 2 t1i + t2i + + tni = (T ∗T )ii = 1: j j j j ··· j j Thus the symplectic group Sp(n) is compact. When n = 1, Sp(1) = q H : q = 1 = t + xi + yj + zk : t2 + x2 + y2 + z2 = 1 : f 2 j j g f g Thus 3 Sp(1) ∼= S : We leave it to the reader to show that there is in fact an isomorphism Sp(1) = SU(2): 424–II 1–6 Although compactness is by far the most important topological property that a group can possess, a second topological property plays a subsidiary but still important role—ˆ connectivity. Recall that the space X is said to be disconnected if it can be partitioned into 2 non-empty open sets: X = U V; U V = : [ \ ; We say that X is connected if it is not disconnected. There is a closely related concept which is more intuitively appealing, but is usually more difficult to work with. We say that X is pathwise-connected if given any 2 points x; y X we can find a path π joining x to y, ie a continuous map 2 π : [0; 1] X ! with π(0) = x; π(1) = y: It is easy to see that pathwise-connected = connected: ) For if X = U V is a disconnection of X, and we choose points u U; v V , then there cannot[ be a path π joining u to v. If there were, then 2 2 1 1 I = π− U π− V [ would be a disconnection of the interval [0; 1]. But it follows from the basic prop- erties of real numbers that the interval is connected. (Suppose I = U V . We may suppose that 0 U. Let [ 2 l = inf x V: 2 Then we get a contradiction whether we assume that x V or x = V .) Actually, for all the groups we deal with the 2 concepts2 of2connected and pathwise-connected will coincide. The reason for this is that all our groups will turn out to be locally euclidean, ie each point has a neighbourhood homeomorphic to the open ball in some euclidean space En. This will become apparent much later when we consider the Lie algebra of a matrix group. We certainly will not assume this result. We mention it merely to point out that you will not go far wrong if you think of a connected space as one in which you can travel from any point to any other, without ‘taking off’.
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