Chapter 1 Linear groups We begin, as we shall end, with the classical groups—those familiar groups of matrices encountered in every branch of mathematics. At the outset, they serve as a library of linear groups, with which to illustrate our theory. Later we shall find that these same groups also serve as the building-blocks for the theory. Definition 1.1 A linear group is a closed subgroup of GL(n; R). Remarks: 1. We could equally well say that: A linear group is a closed subgroup of GL(n; C). For as we shall see shortly, GL(n; C) has an isomorphic im- age as a closed subgroup of GL(2n; R); while conversely it is clear that GL(n; R) can be regarded as a closed subgroup of GL(n; C). 2. By GL(n; k) (where k = R or C) we mean the group of invertible n n matrices, ie × GL(n; k) = T M(n; k) : det T = 0 f 2 6 g where M(n; k) denotes the space of all n n real or complex matrices (according as k = R or C). × 3. We define a norm X on M(n; k) by k k tr(X X) if k = R; X 2 = X 2 = ( 0 X ij tr(X X) k = C; k k i;j j j ∗ if where as usual X0 denotes the transpose and X∗ the conjugate transpose. This is just the usual Euclidean norm, if we identify M(n; k) with kN , where 424–III 1–1 424–III 1–2 2 2 N = n , by taking as the coordinates of the matrix X its n entries Xij. The requirement that a linear group should be closed in GL(n; k) refers to this metric topology. In other words, if T (i) is a sequence of matrices in G tending towards the matrix T GL(n; k), ie 2 T (i) T 0; k − k ! then T must in fact lie in G. Examples: 1. The general linear group GL(n; R) 2. The special linear group SL(n; R) = T GL(n; R) : det(T ) = 1 f 2 g 3. The orthogonal group O(n) = T GL(n; R): T 0T = I f 2 g In other words O(n) = T : Q(T v) = Q(v) v Rn ; f 8 2 g where 2 2 n Q(v) = x + ::: + x (v = (x1; : : : ; xn) R ); 1 n 2 ie O(n) is the subgroup of GL(n; R) leaving the quadratic form Q invariant. 4. The special orthogonal group SO(n) = O(n) SL(n) \ 5. The complex general linear group GL(n; C). This calls for some explana- tion, since GL(n; C) is not a group of real matrices, as required by Defini- tion 1. However, we can represent each complex matrix Z M(n; C) by a real matrix RZ M(2n; R) in the following way. (Compare2 the “realifica- tion” of a representation2 discussed in Part I Chapter 11.) If we “forget” scalar multiplication by non-reals, the complex vector space V = Cn becomes a real vector space RV of twice the dimension, with basis (1; 0;:::; ); (i; 0;:::; 0); (0; 1;:::; 0); (0; i; : : : ; 0);:::; (0; 0;:::; 1); (0; 0; : : : ; i): 424–III 1–3 Moreover each matrix Z M(n; C), ie each linear map 2 Z : V V ! defines a linear map RZ : RV RV; ! ie a matrix RZ M(2n; R). 2 Concretely, in passing from Z to RZ each entry Zj;k = Xj;k + iYj;k is replaced by the 2 2 matrix × Xj;k Yj;k − ! Yj;k Xj;k The map Z RZ : M(n; C) M(2n; R) 7! ! is injective; and it preserves the algebraic structure, ie R(Z + W ) = RZ + RW • R(ZW ) = (RZ)(RW ) • R(aZ) = a(RZ) a R • 8 2 RI = I • R(Z∗) = (RZ)0. • It follows in particular that RZ is invertible if and only if Z is; so R restricts to a map Z RZ : GL(n; C) GL(2n; R): 7! ! Whenever we speak of GL(n; C), or more generally of any group G of complex matrices, as a linear group, it is understood that we refer to the image RG of G under this injection R. The matrix X GL(2n; R) belongs to GL(n; C) if is built out of 2 2 matrices of the2 form × x y ! y− x 424–III 1–4 This can be expressed more neatly as follows. Let 0 1 0 1 1− 0 B C B 0 1 C iI J = B C 7! B − C B 1 0 C B C B .. C @ . A Since any scalar multiple of the identity commutes with all matrices, X M(n; C) = (iI)X = X(iI): 2 ) Applying the operator R, X RM(n; C) = JX = XJ: 2 ) Converseley, if JX = XJ then it is readily verified that X is of the required form. Thus RM(n; C) = X M(2n; R): JX = XJ ; f 2 g and in particular GL(n; C) = T GL(2n; R): JX = XJ f 2 g 6. The complex special linear group SL(n; C) = T GL(n; C) : det T = 1 f 2 g Note that the determinant here must be computed in M(n; C), not in M(2n; R). Thus T = i = SL(1; C); 2 although 0 1 RT = − ! SL(2; R): 1 0 2 7. The unitary group U(n) = T GL(n; C): T ∗T = I f 2 g where T ∗ denotes the complex transpose of T . In other words U(n) = T : H(T v) = H(v) v Cn ; f 8 2 g 424–III 1–5 where H is the Hermitian form 2 2 n H(v) = x1 + ::: + xn (v = (x1; : : : ; xn) C ): j j j j 2 Since R(T ∗) = (RT )0, a complex matrix is unitary if and only if its real counterpart is orthogonal: U(n) = GL(n; C) O(2n) \ 8. The special unitary group SU(n) = U(n) SL(n; C) \ 9. The quaternionic general linear group GL(n; H). The quaternions q = t + xi + yj + zk (t; x; y; z R); 2 with multiplication defined by i2 = j2 = k2 = 1; jk = kj = i; ki = ik = j; ij = ji = k − − − − form a division algebra or skew field. For the product of any quaternion q = t + xi + yj + zk with its conjugate q¯ = t xi yj zk − − − gives its norm-square q 2 =qq ¯ = t2 + x2 + y2 + z2; k k so that if q = 0, 6 q¯ q 1 = : − q 2 k k We can equally well regard H as a 2-dimensional algebra over C, with each quaternion taking the form q = z + wj (z; w C); 2 and multiplication in H being defined by the rules jz =zj; ¯ j2 = 1 − 424–III 1–6 Surprisingly perhaps, the entire apparatus of linear algebra extends almost unchanged to the case of a non-commutative scalar field. Thus we may speak of an n-dimensional vector space W over H, of a linear map t : W W , etc. ! A quaternionic vector space W defines a complex vector space CW by “forgetting” scalar multiplication by non-complex quaternions (ie those in- volving j or k), in just the same way as a complex vector space V defines a real vector space RV . If W has quaternionic dimension n, with basis e1; e2; : : : ; en , then CW has complex dimension 2n, with basis f g e1; je1; e2; je2; : : : ; en; jen : f g Moreover each matrix Q M(n; H), ie each linear map 2 Q : W W ! defines a linear map CQ : CW CW; ! ie a matrix CQ M(2n; C). 2 Concretely, in passing from W to CW each entry Qr;s = Zr;s + iWr;s is replaced by the 2 2 complex matrix × Zr;s Wr;s − ! Wr;s Zr;s The map Q CQ : M(n; H) M(2n; C) 7! ! is injective; and it preserves the algebraic structure, ie C(Q + Q0) = CQ + CQ0 • C(QQ0) = (CQ)(CQ0) • C(aQ) = a(CQ) a C • 8 2 CI = I • C(Q∗) = (CQ)∗. • 424–III 1–7 In this last relation, Q∗ denotes the quaternionic matrix with entries (Q∗)rs = Qsr: To identify M(n; H) as a subspace of M(2n; C), consider the automor- phism of H 1 q q~ = jqj− : 7! In terms of its 2 complex components, q = z + wj q~ =z ¯ +wj: ¯ 7! (Note that q~ =q ¯; indeed, the map q q¯ is not an automorphism of H, but an anti-automorphism.)6 7! Let J denote the diagonal matrix j 0 0 1 J = 0 j M(n; H); B C B .. C 2 @ . A and consider the map 1 Q Q~ = JQJ − : M(n; H) M(n; H): 7! ! We see from above that C(Q~) = CQ; where X¯ denotes (for X M(n; C)) the matrix with entries 2 ¯ Xr;s = Xr;s: Now 0 1 0 1 1− 0 B C B 0 1 C J CJ = B C ; 7! B − C B 1 0 C B C B .. C @ . A which we take the liberty of also denoting by J (as we did earlier, when defining the embedding of GL(n; C) in GL(2n; R), although there we re- garded J as a real matrix rather than a complex one). Thus 1 M(n; H) X M(2n; C): JXJ − = X¯ ⊂ f 2 g 424–III 1–8 1 Conversely, it is readily verified that if JXJ − = X¯, then X is constructed from 2 2 matrices of the form specified above, and so arises from a quater- nionic× matrix.