Hints and Solutions to Selected Exercises

Chapter 1 1.1. Using the formula of Remark 1.4, we obtain

II[~~]II = J2IUI2+1~J4IUI2+1, II [~ ~] II = max{lul, Ivl},

C?S t - sin t] II = 1 cosh t sinh t] II = {t -t} II [smt cost ' II [sinh t cosh t max e ,e . 1.2. (a) From the definition of the matrix norm,

-1 { IBAB-Ixl }{ IAxl } IIBAB II = max Ixl :x t 0 = max IBxl: x t 0

= max {I~~I :x to} = IIAII.

(b) The inequality IICAC- I II ~ IICIlIIAIIIIC- I II always holds.

1.3. (a) We have Ixl 2 = x·x by definition, so

Since the unit sphere {x E en : Ixl = 1} is closed and bounded, the continuous 2 function x I---> IAxl defined on it attains its supremum, hence

2 (b) The matrix A*A is hermitian and for any x E en, x*A*Ax = IAxI , which is real and non-negative; it also vanishes precisely when Ax = O. By the general theory of hermitian matrices, there is an orthonormal basis {UI, ... ,Un} consisting of eigenvalues AI, ... ,An which are all real. For each j,

303 304 Matrix Groups: An Introduction to Lie Group Theory since ujUj = 1; hence Aj ~ O. If A i= 0 there must be at least one non-zero eigenvalue of A*A since otherwise IAxl2 would always vanish; so in this case we have a largest positive eigenvalue A which we might as well assume is A= AI. Now if Ixl = 1 we can 2 2 write x = tIU1 + ... + tnun, where tj EC and ItII + ... + Itn l = 1. Then 2 x* A*Ax = X*(A1tIU1 + ... + AntnUn) = A11tl12 + + An ltnl ~ A11tII 2 + + A11tnl2 = A1, hence IIAII 2 ~ Al and in fact we have equality since uiA*AU1 = AI. 1.4. (a) Let 1 ( . IIAr+111) £ = 2 1 + r~~ lA::lI < 1. For large enough m we have

IIAm + ... + Am+kll ~ IIAmll + ... + II Am+k1l m m m m Am = IIAmll (1 + IIA +l1l + IIA +dIIlA +211 + ... + IIA +lll .. 'II +kll) II Am II IIAmIiIlAm+l1l IIAmll" ·IIAm+k-111

< IIAmll (1 + £ + £2 + ... + £k-l) = II Am II \11-=-£;1-+ I~A_m~1 as k -+ 00.

Also, if mo is large enough, then for m > mo, II~:~~ II < £, so

IIAmll = ::~:~~il·.I!~~:::IIIAmoll < £m-mOIiAmoll-+ 0 as m -+ 00.

Combining these we see that the partial sums of the series form a Cauchy sequence and hence the series converges. (b) Here IIAml1 -+ 00 as m -+ 00 and the series diverges. (c) Any of the usual tests based on absolute convergence works. 1.5. (a) For m, n ~ 1,

m m m m n II~ Arll = IIA + Am+! + ... + Am+nll ~ IIAli + IIAll +! + ... + IIAli +

A n 1 = IIAli m (1 _II ll + ) -+ 0 as m n -+ 00 (1 -IIAII) ,. So the sequence of partial sums is a Cauchy sequence. This can also be done using the ratio test of Exercise 1.4. (b) For any n ~ 1,

(I - A)(I + A + A 2 + ... + An) = 1- An+ 1 = (I + A + A 2 + ... + An)(I - A), so 2 n 1 n II (I - A)(I + A + A + ... + An) - III = IIA + 11 ~ IiAli +l -+ 0 as n -+ 00, and similarly for II(I + A + A 2 + ... + An)(I - A) - III. Hence I - A has inverse

00 r (I - A)-l = LA . r=O Hints and Solutions to Selected Exercises 305

(c) If A k = 0, then

k-1 k-1 (I - A)-l = LAr, exp(A) = L ~Ar. T. r=O r=O 1.6. (a) The subset

O(n) = {A E Mn(R) : ATA - I = O} ~ Mn(R) is bounded since for A E O(n), IIAII = 1. It is also closed since if a sequence Ar E O(n) converges to A EMn(R) then

ATA - I = lim (A;Ar - 1) = lim 0 = O. r-+CX) r-+oo

Therefore O(n) is compact. (b) The subset

U(n) = {A E Mn(C) : A*A - I = O} ~ Mn(C) is compact by a similar argument to (a). (c) Consider the n x n diagonal matrices

A k = diag(k, 11k, 1, ... ,1) (k ~ 1).

Then detAk = 1 so Ak E SLn(k) ~ GLn(k). But IIAkll = k -+ 00 as k -+ 00, so this sequence is unbounded.

1.7. (a) If A, BE H then there are sequences Ar ,Br in H with Ar -+ A and Br -+ B as T -+ 00. Hence ArBr -+ AB and so AB E H since each ArBr E H. Similarly, A;:-l -+ A-I, showing that A-I E H. (b) Here is an alternative proof in terms of open sets which applies to any topological group. Let u, v E H and consider uv. If uv f/. H, the open set G - H contains uv; notice that G - H is the biggest open subset of G which does not intersect H. Since the product map mult is continuous, there is an open set of the form U x V ~ G x G contained in the open set mult-1(G - H) with u E U and v E V. Notice that UnH = 0 = vnH by the above remark. Now take u' E UnH and v' E VnH and note that u'v' = mult(u', v') E H and also mult(u', v') E G - H, giving a contradiction. So uv E H. Also, if wE H, suppose that w-1 E G -H. Then inv-1(G -H) ~ G is open, where inv is the inverse map. But as w E inv-1 (G - H), there must be an element hE H n inv-1 (G - H) and so h-1 E G - H which is impossible. Hence H ~ G. (c) Consider any sequence of diagonal matrices

A k = diag(ak' 1, ... ,1) (k ~ 1) where ak ~ 1 and ak -+ v'2 as k -+ 00 (v'2 could be replaced by any other irrational number greater than 1). Then Ak -+ diag(v'2, 1, ... ,1) f/. r. We must have r = GLn(lR) since for every B = [bij] E GLn(lR) we can find sequences b~7) E Q satisfying b~7) -+ bij as k -+ 00. 1.8. By Proposition 1.26, G ~ Mn(k) is a closed subset of GLn(lR). 1.9. Suppose that G ~ GLn(lR) ~ Mn(lR) for some n. Then H ~ Mn(lR) is closed and bounded since G is compact. For each h E H the set {h} ~ H is open and these form 306 Matrix Groups: An Introduction to Lie Group Theory

an open cover of H by disjoint open sets. By the Hein~Borel Theorem 1.20 there must be finitely many of these sets that cover H, so H is itself finite. 1.10. In each case, we have subgroups Gn+l ~ GLn+l(k) (where k = R or C) and Gn ~ GLn(k) where Gn+! is closed in GLn+l(k) and Gn = Gn+l n GLn(k). As GLn(k) is closed in GLn+l(k), Gn is closed in Gn +!. 1.11. (a) If A E SymP2m(R), then ATJ2mA = J2m and as detJ2m = 1, we have detA2 = 1, so detA = ±1. (b) Let A = [~ ~] E SymP2(R). By definition, [~ ~][_~ ~][~ ~]=[_adO+bC adobc]=[_~ ~], so A E SymP2(R) if and only if detA = Ii hence Symp2(R) ~ SL2(R). 1.12. Clearly Pn U(n) ~ pn GLn(C). If Z = [Zr.] E Mn(C) where Zr. = Xr• +Yr.i with Xr.,Yr. E R, then Pn(Z·) = Pn([X.r] - [Y.riJ) = Pn(Z)T. Hence Z E U(n) if and only if Pn(Z) E O(2n). If A E O(2n), then A E SymP2n(R.) if and only if Ahn = hnA and by Proposition 1.44 this is equivalent to A E pn U(n). This shows that pn U(n) = O(2n)nSymP2n(R).1f BE Symp2n(JR) then Bhn = hnB T if and only if B = B. This shows that pn GLn(C)nSymP2n(R) = O(2n)nSymP2n(R). The second collection of equalities is proved in a similar way. 1.13. (a) The triangle inequality for P follows from the following sequence of inequal• ities valid for any (XI,X2), (YI,Y2), (ZI,Z2) E Xl X X2,

P((XI, X2), (Zl, Z2)) =JPI (Xl, ZI)2 + P2(X2, Z2)2 ~J(PI(XI' YI) + PI(YI, zdF + (p2(X2, Y2) + P2(Y2, Z2))2

[by the triangle inequalities for PI and P2]

[by the usual triangle inequality for R2]

The other properties required for p to be a metric are straightforward to check. If UI x U2 ~ Xl X X2 with UI ~ Xl and U2 ~ X2 open, then for each Xl E UI and X2 E U2 there are open discs NX1>Pl(Xlirl) ~ UI and NX2,P2(X2ir2) ~ U2i for r = min{rl,r2}, we have NX1XX2,P((XI,x2)jr) ~ UI x U2, so UI X U2 is open with respect to p. Conversely, for an open disc NX1XX2,P((XI,x2);r) we have

NX1,Pl(xl;r/v'2) x NX2,P2 (X2i r/v'2) ~ NX1XX2'P((XI,x2)jr). Hence every subset of Xl x X2 open with respect to P is a union of products of open discs in Xl and X 2 • So the topology associated with p has the same open sets as the product topology. (b) If (XI,r,X2,r) -. (XI,X2) say, then for i = 1,2, Pi(Xi,r, Xi) ~ P((XI,r,X2,r), (XI,X2)), Hints and Solutions to Selected Exercises 307

SO Xi,r ...... Xi as r ...... 00. Conversely, if XI,r ...... Xl and X2,r ...... X2 then

p((XI,r, X2,r), (Xl, X2» = viPI (XI,r, XI)2 + P2(X2,r, X2)2 ...... 0, so (XI,r, X2,r) ...... (Xl, X2) as r ...... 00. 1.14. (a) If 9 tJ. Staba(x) then gx E X - {x}. Since X - {x} is an open subset of X, p,-I(X - {x}) ~ G x X is open, so there are open sets U ~ G, V ~ X with 9 E U, x E V and U x V ~ p,-I(X - {x}). But then for hE U we must have hx =I x, hence U ~ G- Staba(x). This shows that G- Staba(x) is open in G. (b) If 9 tJ. Staba(W) then gW i. W or g-IW i. W. If gW i. W then for some w E W, gw =I w. Now a similar argument to (a) shows that there is an open set U ~ G- Staba(W) containing g. If g-IW i. W then we find an open set U' ~ G- Staba(W) containing g-I; applying the inverse map inv we obtain an open set invU' ~ G-Staba(W) containing g. Finally, since each stabiliser Staba(w) is closed the intersection of any collection of such stabilisers is also closed.

1.15. (a) Taking PI to be the norm metric on Mn(k) and P2 to be the usual metric on kn, define p as in Exercise 1.13, i.e., set p(A,x) = vlIIAII 2 + \xI 2. If A,B E Mn(k) n and x,y E k , then lAx - Byl = I(Ax - Bx) + (Bx - By)1 ~ IIA - Blllxl + liB II Ix - yl· A routine £;-8 argument now establishes continuity of the product map at each (A, x). (b) These are consequences of results in the previous exercise which show that these stabiliser subgroups are closed. (c) We have {[~ StabGLn(R)(en) = On1I,I]: A E GLn-I(IR), BE MI,n-I(IR)},

StabGLn(R)(X) = {[~ OntI,I]: A EGLn-I(IR), BE MI,n-I(IR), t E1R}, {[~ StabSLn(R) (en) = On1I'I]: A E SLn-I(IR), BE MI,n-I(R.)},

StabSLn(R)(X) = {[~ ~;1~II]: A E GLn-I(R.), BE MI,n-I(R.)}, OI~-I Stabo(n)(en) = {[ On1I'I]: A E O(n - 1), },

Stabo(n)(X) = {[OI~-I OntI'I]: A E O(n - 1), t = ±1},

Stabso(n)(en) = {[OI~-I On1I'I]: A ESO(n -I),},

StabsO(n) (X) = {[OI~-I OntI,I]: A E O(n - 1), t = detA}.

1.16. (b) We have [~ ~]xn=(ax+cyr,

so StabSU(2)(Xn) = {[~ ~]: un = I}. 308 Matrix Groups: An Introduction to Lie Group Theory

(c) We have

2 [~ ~] xy = (ax + cy)(bx + dy) = abx + (ad + bc)xy + cdy2, so StabsU(2)(XY)={[~ ~] :lul=I}U{[~v ~] :!VI=I}. n (d) If A E kerrpn, then A E Stabsu(2)(x ), so by (b),

Also, for each T = 1, ... , n,

2r 2 so this can only happen if u = 1 for all such Tj hence u = 1. If n is odd, this gives u = 1, while if n is even u = ±1. 1.17. (a) Arguing as in the discussion of the norm of a square matrix at the beginning of Section 1.2, there is a number K ~ 0 for which satisfies v'(rp(v)) ~ K v(v) for every v E V. Thus for x,y E V we have

v'(rp(y) - rp(x)) = v'(rp(y - x)) ~ Kv(y - x) -+ 0 as y -+ x, hence rp is continuous. (b) These projections are linear transformations, hence by (a) they are continuous. Now ker rp' = Wand so W = rp,-1 {O} is a closed subset of V. Chapter 2 2.1. For n ~ 0 we have

hence

00 e ~ (_I)n n+l] (2n+ I)! _ [cost sin t] 00 (_I)nt2n -- sint cost· ~ (2n)!

Similarly, 0 1] 2n 0 1] 2n+1 _ [0 1] [1 0 =h, [1 0 - 1 0' hence 2n 1 ~00 t + ] t]) = ~ ~ (~:)! (2n + I)! _ [COSh t sinht] o 00 en+1 00 t2n - sinht cosht . ~ (2n+ I)! ~ (2n)! Hints and Solutions to Selected Exercises 309

Finally, by an easy induction on n,

n to] n [ t [-2 t = _2ntn- 1 hence

n

00 t ] t ° _ "'-~ n! ° eXP([_2 t]) - 00 ntn-1 00 tn [ -2L --;;J L n! n=O r=O 2.2. (a) Each partial sum has the form

~ ~(BAB-1r = B(~~Ar) B-1.

(b) Take A = D and B = C in (a) to obtain 1 l exp(D) = C exp(diag(Al,"" An ))C- = C diag(e>'l, ... ,e>'n )C- .

(c) The eigenvalues of [_~ ~] are ±ti and the matrix C = [~ _~] satisfies

l [_~ ~] = C diag(ti, -ti)C- .

ti ti Now using the identities cost = (e + e- )/2 and sint = (e ti - e-ti )/2i we obtain

exp Cd' (ti -ti)C-1 _[ cost sint] ([_Ot °t]) -_ lag e ,e -_ sin t cost . Similarly, 1 [~ ~] = C diag(t, _t)C- ,

t t t where C = [i -n· Using the identities cosh t = (e + e- )/2 and sinh t = (e • e-t )/2 we obtain 0 t]) _Cd' (t -t)C-1 _ [coSht sinht] exp ([t ° - lag e , e - sinh t cosh t . 2.3. (a) Every positive power of N is strictly upper triangular and N is nilpotent, say N k = O. So exp(N) is upper triangular with l's down its main diagonal. (b) Write N = tl + U, where t E k and U is strictly upper triangular. Then U is nilpotent, say Ui for some £ > 0. So for m ~ 1 we have

For m ~ £ this becomes 310 Matrix Groups: An Introduction to Lie Group Theory

Thus

r exp(N) = ~ ~! min~,m} (;)tm-ru = ~ (~ ~! (;)tm-r) Ur.

2.4. (a) The operation of transposition is a continuous function on Mn(JR) so it com• mutes with taking limits of sequences. Alternatively, the partial sums of exp(S) satisfy

(~~!Sk)T = ~ ~! (ST)k = ~ ~! (_S)k = exp(-S) = exp(S)-I,

hence exp(Sf = exp(S)-I. (b) Similarly, the hermitian conjugate ofthe exponential satisfies exp(S)* = exp(S)-I. 2.5. (a) Start by solving the differential equation

o I (t) = o(t) [-10 0(0) = I. -2]1 ' The solution is

The desired solution is t X(t)] = [e- [y(t) 0

(b) The solutions are

[:m] = [e-t(;:s-tt~~i~t)]' [:m] = [:=:J. 2.6. A solution has the form x(t) = exp(tA)x(O) for t ER Since A* = -A we have exp(tA)T = exp(-tA) = exp(tA)-I, hence exp(tA) is orthogonal. So

2 2 Ix(t)1 = x(t) . x(t) = (exp(tA)x(O» . (exp(tA)x(O» = x(O) . x(O) = Ix(0)1 • We also have x(t) . x'(t) = ~ : t Ix(t)1 2 = 0, showing that x(t) and x/(t) are orthogonal. 2.7. (a) Choose any sequence of matrices 1 o o

o >'r-I,n 1 o o >'r,n in which the r diagonal >.'s in J(>', r) have been replaced by r sequences of non-zero terms >'I,n, ... ,>'r,n that satisfy

>'j,n -+ >. as n -+ 00. Hints and Solutions to Selected Exercises 311

Then each matrix An has r distinct eigenvalues so is diagonalisable and An ---+ J(>', r) as n ---+ 00. (b) This follows from (a) together with Theorem 2.9. Chapter 3 3.1. (a) Any abelian Lie algebras of the same dimension are isomorphic since they are isomorphic as vector spaces. So let g be any 2-dimensional k-Lie algebra which is not abelian. Then there are elements x, y E g for which [x, y] # 0; these elements cannot be linearly dependent so they form a basis of the k-vector space g. If [x, y] = rx + sy with r, s E k, we can interchange x, y if necessary to ensure that r # 0, and then replace x by (r-1)x to ensure that [x, y] = x + sy. Finally, since [x + sy, y] = x + sy, we can replace x by x + sy to ensure that [x, y] = x. Notice that in b we have the elements U= [0 0'1] V= [-100] 0 ' which have the bracket [U, V] = U. Then there is an obvious isomorphism of Lie algebras g --+ b under which x <---> U and y <---> V. (b) Take G={[~ ~]}~GL2(k).

3.2. (a) If "{: (a, b) --+ G is a differentiable curve then so are the curves defined by (U"{)(t) = U"{(t), ("{U)(t) = "((t)U, (U"{U-1)(t) = U"{(t)U-I, and these have derivatives 1 (U"{)'(t) = U"{'(t), ("{U)'(t) = "{'(t)U, (U"{U-1)'(t) = U"{'(t)U- . (b) Use the derivative maps at I, i.e., >'u = dLu, pu = dRu and XU = dCu. The required properties follow easily. 3.3. Follow the approach of Section 3.3. The Lie algebra of G 1 is gl={[: ~]:U,V,WER}.

The elements U = [~ ~], V = [~ ~] and W = [~ ~] form a basis with the two non-trivial brackets [U, W] = -Wand [V, W] = W. The Lie algebra of G2 is

and this is abelian. The Lie algebra of G3 is ~ ~ ~ .. {[_: ] r, " t, _, v, W E + The Lie algebra of G4 is n g4={[~ ~] :AEMn(k), tEk }, 312 Matrix Groups: An Introduction to lie Group Theory

with Lie bracket given by

Consider the 2m x 2m real matrix Au A2I A= . [ AmI

where each Ar• EM2 (R). Then AThm + J2mA = 0 if and only if the equations

A;rJ + JAr. = 0 (r,s = 1, ... ,m) are satisfied and these are equivalent to

A;r = JAr.J (r,s = 1, ... ,m).

If we write An = [~; ~:] then

rr If s = r, this gives Arr = [arr b ]. Crr -arr

3.4. We have b S! a/ ker <1>. Then b is abelian if and only if for every commutator [x, vj in a, [x, vj + ker = 0 + ker <1>, i.e., [x, vj E <1>. Hence b is abelian if and only if a ~ ker. 3.5. If exp(sX) exp(tY) = exp(tY) exp(sX) for all s, t E JR, then d d [X, Yj = -d -d (exp(sX) exp(tY) - exp(tY) exp(sX)) = O. s 1.=0 t 1.=0 Conversely, if [X, Yj = 0 then for each s E R, consider the function

-y: R --+ G; ')'(t) = exp(sX) exp(tY) exp(-sX).

Then ')'(0) = I and

')" (t) = exp(sX)Y exp(tY) exp(-sX) = exp(sX) exp(tY) exp(-sX)Y = ')'(t)Y since exp(sX)Y exp(-sX) = Y by definition of the exponential function. But the differential equation ')"(t) = ')'(t)Y, ')'(0) = I has the unique solution ')'(t) = exp(tY), so for all s, t we have exp(sX) exp(tY) exp(-sX) = exp(tY) and therefore exp(sX) exp(tY) = exp(tY) exp(sX). Hints and Solutions to Selected Exercises 313

3.6. (a) If A E U then

(A)T = (I - AT)(I + AT)-l = (I - A-I)(I + A-I)-l =(A - I) (A + I)-I = -(A), so (A) E Sk-Symn(lR). Conversely, if BE Sk-Symn(lR) then

((I - B)(I + B)-I) = (I - (I - B)(I + B)-I) (I + (I - B)(I + B)-l)-l = (I - (I - B)(I + B)-I) (I + (I - B)(I + B)-l)-l = ((I + B) - (I - B)) ((I + B) + (I - B))-l = (2B)(2I)-1 = B,

and ((I - B)(I + B)-I)T (I - B)(I + B)-l = (I - BT)(I + BT)-I(I - B)(I + B)-l = (I + B)(I - B)-I(I - B)(I + B)-l = I.

Hence (I - B)(I + B)-l E O(n) and <1>((1 - B)(I + B)-I) = B. In fact, there is a path [0,1) -+ O(n) given by

t 1-----+ (I - tB)(I + tB)-I, hence det(I-tB)(I+tB)-1 = 1 since detC = ±1 if CE O(n). So (I - B)(I+B)-l E SO(n) and therefore im = Sk-Srmn (lR). (b) By the calculation in (a), <1>- (B) = (I - B)(I + B)-I. (c) We have

dimSO(n) = dimSk-Symn(R) = 1 + 2 + ... + (n - 1) = (;).

3.7. Parts (a)-(c) are very similar to the previous question and we obtain

8-I (B) = (I - B)(I + B)-l (B E Sk-Herm2(C)), and 2 dim D(n) = Sk-Hermn(C) = n + 2 (;) = n .

(d) In the case n = 2, if A E V n SD(2), then the eigenvalues of A must have the form >., X EC where 1>'1 = 1. Then the eigenvalues of 8(A) are (1 - >')/(1 + >.) and (1 - X)/(1 + X), hence

I-A I-X I-A A-I tr8(A) = 1 + A + 1 + X = 1 + A + A+ 1 = O.

Conversely, if BE Sk-Hermg(C) then B has imaginary eigenvalues ±ti for some t E R, so 8-I (B) has eigenvalues (1 - ti)/(1 + ti) and (1 + ti)/(1 - ti). Thus we have

det 8-1 (B) = (1 - ti) (1 + ti) = 1. (1 + ti) (1 - ti) 314 Matrix Groups: An Introduction to Lie Group Theory

Examples show that this can be false when n > 2.

3.8. (a) The surjection det: O(n) ---+ {1, -I} has kernel SO(n). The diagonal matrix diag(I, ... , 1, -1) E O(n) generates a subgroup C = {I, diag(I, ... , 1, -I)} of order 2 for which O(n) = C SO(n), en SO(n) = {f}. (b) Let 1l" = {diag(I, ... ,I,z) E U(n): Izl = 1}:::; U(n). Then U(n) = 1l" SU(n), '[" n SU(n) = {f}. (c) Let D = {diag(I, ... ,1, t) E GLn(JR.) : t E JR.} :::; GLn(JR.). Then GLn(JR.) = D SLn(JR.), D n SLn(JR.) = {f}. (d) Let D' = {diag(I, ... , 1, z) E GLn(C) : t E C} :::; GLn(C). Then GLn(C) = D' SLn(C), D' n SLn(C) = {f}.

Chapter 4 4.1. Some elements of order 2 in A~ are -1 and 1 e3(1 2) = 6[ 2(12) - (1 3) - (23) ], 1 e3(1 3) = 6[ 2(13) - (23) - (12)], 1 e3(2 3) = 6[ 2(2 3) - (1 2) - (1 3»).

As JH[ has only the element -1 of order 2 these algebras cannot be isomorphic. 4.2. (a) The subset Af ~ A is closed and bounded, so compact. (b) Let u, v E Af. Then we have the inequalities v(uv) :::;v(u)v(v) = 1, I I V«UV)-l) = v(V-1U- ) :::;v(v-I)V(U- ):::; 1, 1 = 1.1(1) = v(uV(UV)-I) :::;v(uv)V«UV)-I) :::; 1, hence v(uv) = v((UV)-I) = 1. So Af :::; A x. (c) Cf is the unit circle with abelian Lie algebra {ti : t E lR.} :::; C.

JH[f ={u+vj:u,vEC, luI 2 +lvI2 =1}, with Lie algebra

{ri + sj + tk: u,v E C, r,s,t E JR.} ~ .5u(2).

For k = JR. or C, let A = Mn(k)f. Suppose that IAul < lui for some u E knj then

1 = M = IA-1(Au)1 < IA-1(Au)1 ~ IIA-1 1I lui lui IAul '<: , Hints and Solutions to Selected Exercises 315

n contradicting the fact that IIA-III = 1. This shows that for every x E k , IAxl = Ixl. By Proposition 1.38 and the analogous result for unitary groups, this shows that A E O(n) if k = JR and A E U(n) if k = e, while the corresponding Lie algebras are o(n) ~ Mn(JR) and u(n) ~ Mn(C). 4.3. For the case r > 0, let q E JH[ satisfy l = r. Using notation from the proof of 2 Proposition 4.40, we find that JR(q) is a field, so the number of roots of x - r is at most 2 and these are the real numbers ±JT. The case where r < 0 can be dealt with using Proposition 4.42.

4.4. Here is what happens when n = 1. For JH[R, A: JH[ --+ M4(JR) takes the values A(l) = /4 and ~ ~ -~ [~ ~ -~ ~], [~ ~ -~ A(i) !J] ,A(j) = A(k) = o 0 -1 0 0 1 0 0 and so

A(a1 + bi + cj + dk) = [ad~ -; =~ =~. -c b ~J The reduced determinant RdetR: JH[x --+ JR x takes the value

RdetR(a1 + bi + cj + dk) = det A(a1 + bi + cj + dk) = a4 + 2a2 b2 + 2d2 a2 + 2c2 a2 + b4 + 2d2 b2 + 2c2 b2 + c4 + 2c2 d2 +~.

For JHJ:c, A: lHI --+ M 2 (C) has the effect

A(l) = h A(j) = [~ -~],

so A(a+b]). = [ab -b]a' x When n = 1, the reduced determinant Rdetc: lHI --+ ex takes the value

2 2 Rdetc(a1 + bj) = det A(a1 + bj) = lal + Ib1 . 4.5. (b) We have

exp(r + su) = exp(r)exp(su) = er(coss 1 + sinsu).

4.6. (a) g = Mn(lHI). (b) g = {A E Mn(lHI) : A* = -A}. (c) When n = 1, taking k = JR and using Exercise 4.4, we have a1 + bi + cj + dk EG with a, b, c, d E JR if and only if a4 + 2a2 b2 + 2d2 a2 + 2c2 a2 + b4 + 2d2 b2 + 2c2 b2 + c4 + 2c2 d2 + ~ = 1, so 9 = {ri + sj + tk : r, s, t E JR}. For k = e, u1 + vj EG with u, vEe if and only if lul 2 + Ivl 2 = 1, giving

9 = {ri + zj : r E JR, z E C}. 316 Matrix Groups: An Introduction to Lie Group Theory

(d) When n = 1,

2 2 2 2 Sp(l) = {al + bi +cj +dk: a,b,c,d E R, a + b +c +d = 1}, so again using Exercise 4.4, for k = R or C we obtain

9 = {ri + zj : r E R., z E q.

Chapter 5 5.1. (a) First write u = rIel + ... + rnen with rk E R and r~ + ... + r~ = 1. Then

n Also, if wE R. with w· u = 0, then writing w = Slel + ... + Snen with Sk E R, we have wu = (rlsle~ + ... + rnS~e~) + L (riSj - rjSi)eiej l";i

A similar calculation shows that

uw = L (rjSi - riSj)eiej = -wu. l";i

(c) The elements Ael,"" Aen form an orthonormal basis of Rn ~ GIn, so using n the universal property of Theorem 5.4, the function R ----+ GIn sending :E~=l Xiei to :E~=l xiAei induces a homomorphism A.: GIn ----+ GIn. The inverse A-I = AT induces (A-I).: GIn ----+ GIn which is actually the inverse of A•. A straightforward calculation shows that A.(el ... en) = (det A)el ... en. i~: 5.2. (a) The image of GIn ----+ Gln+l is spanned by the monomials eil '" ei r with ik = 1, ... ,no (b) Since x . en+l = 0,

2 2 (xen+l)2 = xen+lxen+l = -x2e~+l = -(-lxI )(-I) = -lxI , so the universal property gives the desired homomorphism. Also notice that xen+l E GI;t and it is easy to see that every monomial eil ... ei2. of even length with ik = 1, ... , n + 1 is in the image. 5.3. (a) For any k = 1, ... , n, while Hints and Solutions to Selected Exercises 317

Since n - k +1 == k mod 2, ekWn = wnek. From this it follows that for every monomial in the ek,

(b) We have W~ (_1)(n-1)+"'+le~ e~ 1 = e1 .•. ene1 ... en = ... = (_l)(;)+n = (_1)(nt ).

But (n + 1) == 0 mod 4, so (n~l) == 0 mod 2 and therefore w~ = 1. Now

2 1 2 1 ) O± = 4(1 ± 2Wn +wn ) = 4(2±2wn = O±, so these are idempotents which are central and satisfy 0+ + 0_ = 1. (c) Set O+A+ = Cln and O_A_ = Cln. (d) By Exercise 5.2, CIt = Ch = lBI and Cli = Cl6 = Ms(R). In each case multiplica• tion by O± maps the simple algebra CIt onto A± and therefore gives an isomorphism CIt ~ A±.

5.4. (c) Send "( to -1 and ej to ej. (d) The elements (1/2)(1 ± "() E JR[CIGPnl are central idempotents. Put A = (1/2)(1 + "()JR[CIGPnJ, B = (1/2)(1 - "()R[CIGPnl. 5.6. (b) Z(O(n)) = Z(SO(n)). Chapter 6 6.1. (b) For some Q E Lor(3, 1), cost - sint o _ Q sint cost o o ] -1 At - 0 0 [ cosh(-t) sinh(-t) Q. o 0 sinh(-t) cosh(-t) (c) With the same Q as in (b), take U~Q~ -~ j _~Q-'

(d) A suitable equation is o'(t) = o(t)U, 0(0) = 1. 6.2. (a) We have ~d& ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ + ij = 318 Matrix Groups: An Introduction to Lie Group Theory and 1 0 0 0 cosh..;2t + 2 cosh..;2t - 2 sinh ..;2t 0 2 2 ..;2 a(t) = exp(t(P24 + P34 )) = cosh..;2t - 2 cosh..;2t + 2 sinh ..;2t 0 2 2 ..;2 sinh ..;2t sinh ..;2t cosh ..;2t 0 ..;2 ..;2 2 (b) We have ~(10- l -irE+ F) ~ [1 0 i ] 2(1 + i) and

cosh(t/..;2) sinh(t/..;2) (1 _ i)] ..;2 . a(t) = exp(-ti(E + P)) = sinh~..;2) (1 + i) [ cosh(t/ ..;2)

(c) The homomorphism Ad: SL2(C) ----+ Lor(3, 1) relates these two curves through the equation a = Ad 0 a. Chapter 7 7.1. (a) Define the function

P: Mn(lR) x IR ----+ IRj F(A, b) = bdetA-1. This is smooth and M = p-10 ~ Mn(lR) x IR is a closed subset. The derivative mapping d = d(A,b) at a point (A,b) E p-10 has the form d(X,v) = (detA)v + ddetA(X), where we make the natural identifications T(A,b) Mn(lR) x IR = Mn(lR) x IR and T t IR = IR for any t E R. Since det A =I 0, d is surjective at such a point. Hence M ~ Mn (1R) x IR 2 is a smooth submanifold of dimension equal to dim Mn(lR) = n • We also have

T(A,b) M = {(X, (detA)-l ddetA(X)) : X E Mn(IR)}.

(b) We have M £':! GLn(IR). 7.4. In each case we have to show that the subgroup is closed in G. In (a-c) we also use the continuous action (as introduced in Section 1.9, see the Exercises of Chapter 1 for details), 1 J.L: G x G ----+ G; J.L(x,y) = xyx- • (a) Za(g) agrees with the stabiliser of g, Staba(g) which is a closed subgroup of G. (b) Z(G) is the intersection of all the closed sets Za(g) for 9 E G and so is closed. (c) Na(g) is identical with the generalised stabiliser Staba(H) ~ G, which is a closed subgroup. (d) Let x E G- kerc,o. Then c,o(x) =11, so there is an open subset U ~ H containing Hints and Solutions to Selected Exercises 319

rp(x) but not 1. Then rp-1U S; G is open since rp is continuous. Clearly kerrpnrp-1U = {(If and x E rp-1U. This shows that G - kerrp is open in G, hence kerrp is closed. 7.5. This is similar to the previous exercise.

7.6. Use the smooth maps Lg, Rg and Xg of Proposition 7.16. The tangent spaces are the images of the derivatives at 9 applied to the tangent space of H.

7.7. By Exercise 7.4(d), kerrp ~ G is closed. For 9 E kerrp we have T g kerrp = kerdrpg. 7.8. (a) Use the continuous function

for which e2n = P-10. (b) The eigenvalues of an element J E e2n are ±i. Since the minimal polynomial of J over C is clearly X 2 + 1, which has no repeated roots, J is diagonalisable over C, and over lR has the form h 0 0

0 J2 0 1 J=B B- , where B E GL2n(lR) and h = [~ -1]o . h 0 0 0 h

(c) These stabilisers are the subgroups Pn GLn(C) and P~ GLn(C). (d) For techniques useful for showing that e2n is a submanifold, see Chapter 8. The tangent space to e2n at J can be identified with

kerdPJ = {X E M2n (lR) : XJ + JX = O}.

Chapter 8

8.2. (a) The obvious quotient homomorphism Sp(n) x Sp(1) ----+ Sp(n) xSp(l) has the discrete subgroup {(In, 1), (-In, -I)} as its kernel, so the derivative homomorphism is an isomorphism of Lie algebras; the Lie algebra of the domain is the direct product sp(n) x sp(I). (b) We can identify JH[Pn-1 with (Sp(n) xSp(I))/(Sp(n -1)x Sp(I)). Chapter 9

9.1. (a) For each connected component U of G, choose an element 'U E U. G has a countable open covering {U1 ,U2 , •••}, hence for each 'U as above there is an open set Uiu S; U containing 'U. This means that the components form a countable set. (b) The components form an open covering of G. By the Heine-Borel Theorem 1.20, a finite subcollection also covers G. As these sets are disjoint there must be only finitely many of them. 9.2. (a) We have

StabsL,,(R)(en) = {[~ OnI1'1]: A E SLn-1(R), BE M1,n-1(lR)},

n OrbsL,,(R)(en) = lR - {O}. 320 Matrix Groups: An Introduction to Lie Group Theory

n (b) This homogeneous space can be identified with OrbsLn(R.)(en) = R - {O} and this is path connected since every pair of vectors u, v E R.n - {O} which are not parallel can be joined by a path of form t ~ tu + (1 - t)v (t E [0,1]). Now use induction on n as in Example 9.14 to show that SLn(R) is path connected. 9.3. (a) We have

StabSLn(c) (en) = {[~ Oni1'1]: A E SLn-l(C), BE Ml,n-l(C)} , n OrbSLn(c)(en) = C - {O}. (b) This is similar to part (b) of the previous exercise. (c) Since GLn(C)j SLn(C) is identified with C X using the determinant, Proposi• tion 9.11 shows that GLn(C) is path connected. 9.4. (a) Ifx E an, then x T AATx = IATxl 2 ~ 0, and this vanishes only when ATx = 0, which happens only when x = 0 since AT is invertible. The eigenvalues of S are then positive real numbers and S is diagonalisable, say S = Pdiag(Al, ... , An)pT for some T P E O(n) and Ai> O. Then we take Sl = Pdiag(~, ... , A)p . (b) We have (SllA?(SllA) = AT(Sf}-lSllA = ATS-1A = AT(AAT)-lA = AT(AT)-lA-IA = I.

(c) Since RRT = I = SST, we have

(d) S2 must have the same eigenvalues as diag(~, ...,A) and indeed

S2 = U diag(~, ... , .J>:::)UT for some U E O(n). So

U diag(Al"'" An)UT = diag(Al,"" An) and hence

By comparing entries we find that this is only possible if either Ai = Aj for all i,i or U is diagonal, in which case S2 = diag(~, ... , A). (e) Use (c) and (d). (f) First show that the set of positive definite symmetric matrices is a path con• nected subset of GLn(R.) by finding a path from every such matrix of the form V diag(Al, ... , An}VT with V E O(n) to I. 9.5. (a) We have

n StabAffn(k) (0) = {[~ ~]: A E GLn(k)}, OrbAffn(k) (0) = k . Hints and Solutions to Selected Exercises 321

Notice that StabAffn{k)(O) can be identified with GLn(k). (b) This follows from the corresponding facts for GLn(k). 9.6. See Section 8.5 for background on this. The spaces in (a) and (b) are connected and actually diffeomorphic. 9.7. This is similar to the previous exercise; the homogeneous spaces are diffeomorphic and connected.

9.8. (b) The Pfaffian function pf: E 2m ---+ R X is surjective, so E2m has at least two path components. In fact it has exactly two components since the diffeomorphic homogeneous space GL2m(R)/ SymP2m(R) has at most two components; this implies that SymP2m(lR) ~ SL2m(R). 9.10. By straightforward calculation we find that

NU(2)(T) = T U {[~ ~]: lui = Ivl = I} ~ U(2).

This has the two path components T, {[~ ~]: lui = Ivl = I}, so 7l"o NU(2)(T) ~ {I, -I}. 9.11. Using the previous exercise, we obtain

which has 4 path components each of which contains one of the matrices

where h = [~ -~]. Then 7l"o Na(T') ~ {I, -I} x {I, -I}.

Chapter 10 10.1. Given a Lie homomorphism ep: T ---+ T, the curve "1: R ---+ T given by 'Y(t) = ep(e2"it) is a one-parameter subgroup satisfying 'Y'(s + t) = 'Y(sh'(t) for all s, t E IR. By our general results on such curves we have 'Y(t) = e2"cit for some c E R. But whenever nEZ, 'Y(n) = ep(l) = 1, so c E Z. For ep to be an isomorphism we must have c = ±1. For r ~ 1, for each integer matrix A = [aii] E Mr(Z), there is a Lie homomorphism with derivative d = A acting in the obvious way on t = Ri x ... x Ri. Isomorphisms come from those A with integer inverse A-I E Mr(Z). 10.2. IfTo = T is not maximal, then it is contained in another TI of greater dimension. Repeating this, for dimensional reasons eventually we obtain a sequence of tori To ~ T I ~ ... ~ Tk ~ G where there is no torus in G which properly contains Tk. 10.3. (a) We have

NU(2)(T)=TU{[~ ~] :IUI=IVI=I}=TU[~ ~]T, 7l"oNu(2)(T)={I,-I}. 322 Matrix Groups: An Introduction to lie Group Theory

The conjugation action is given by

[~ ~] T· diag(z, w) = diag(w, z).

(b) and (c) are discussed in Chapters 11 and 12. 10.4. See Chapters 11 and 12. 10.5. (c) O(n) is not connected. Chapter 11 11.1. (a) By Theorem 11.5, z(u(n)) consists of all scalar hermitian matrices tiIn with t E lR. Using the defining Equation (11.1), we have XE z(u(n)).L if and only if tr(tiX) = 0 for all t E R. Since tr(tiX) = titr(X), this gives

z(u(n)).L = {X E u(n) : tr X = O} = $U(n).

(b) Compose the determinant det: U(n) --+ T ~ Z(U(n)) with the projection Z(U(n)) --+ Z(U(n))/Z for Z = {ZIN : zn = I}. (c) Define this homomorphism by (A, B) 1----+ AB. 11.4. See Chapter 12 for details. 11.5. (a) Take Tl = {diag(u,v,w,uvw) : lui = Ivl = Iwl = I}. Then 2 NG1(Tl)={[0~2 011 ] :A,BENU (2)(T), detAdetB=l}, where NU(2)(T) ~ U(2) is the normaliser of the maximal torus used in Exercise 10.3. NG1 (TI) has 4 components and the Weyl group W Gl (Tl ) = NGl (Td/Tl consists of the cosets of the matrices

2 2 2 2 1 WI = [1 0 ,2] W2 = [h 0 • ] Wg = [ h 0 2.2] 4, 02.2 J2 ' 02.2 12' 02,2 J2 ' where J2 = [~ -~]j then WG1(Tl) ~ {1,-1} x {1,-1}. WG1(Tl) acts on Tl by

WlTl . diag(u, v, w, uvw) = diag(u, v, ffiiW, w), W2Tl . diag(u, v, w, uvw) = diag(v, u, w, ffiiW), WgTl . diag(u, v, w, ffiiW) = diag(v, u, ffiiW, w).

The Lie algebra of Tl is tl = {diag(ri, si, ti, -(r+ s + t)i) : r, s, t E R}, and there are 4 non-trivial roots ±Ol, ±02, where

01 (ri, si, ti, -(r+ s + t)i) = r - s, 02(ri, si, ti, -(r + s + t)i) = r + s + 2t, for r, s, t E R. W Gl (Tl) acts on the roots by

WlTl • 01 = 01, W 2T l • 01 = -01, WgTl . 01 = -01, WlTl . 02 = -02, W 2T l • 02 = 02, WgTl . 02 = -02. (b) This is similar to (a), see also Exercise 9.11. Chapter 12 12.2. Rotate and rescale Figure 12.3 into Figure 12.4. Bibliography

[1] J.F. Adams, Lectures on Lie Groups, University of Chicago Press (1969). [2] J.F. Adams, Lectures on Exceptional Lie Groups, University of Chicago Press (1996). [3] M.F. Atiyah, R. Bott & A. Shapiro, Clifford modules, Topology 3 (1964) suppl. 1, 3-38. [4] R. Carter, G. Segal & LG. Macdonald, Lectures on Lie groups and Lie algebras, Cambridge University Press (1995). [5] P.M. Cohn, Classic Algebra, Wiley (2000). [6] 1. Conlon, Differentiable Manifolds: A First Course, Birkhauser (1993). [7] M.L. Curtis, Matrix Groups, Springer-Verlag (1984). [8] R.W.R. Darling, Differential Forms and Connections, Cambridge Univer• sity Press (1994). [9] A. Dold, Lectures on Algebraic Topology, Springer-Verlag (1995). [10] W. Ebeling, Lattices and Codes, Vieweg (1994). [11] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, American Mathematical Society (2001). [12] R. Howe, Very basic Lie theory, American Mathematical Monthly 90 (1983), 600-623; Correction, Amer. Math. Monthly 91 (1984), 247. [13] W.Y. Hsiang, Lectures on Lie Groups, World Scientific (2000). [14] V.G. Kac, Infinite-dimensional Lie Algebras, Cambridge University Press (1990).

323 324 Matrix Groups: An Introduction to Lie Group Theory

[15] S. Lang, Algebra, second edition, Addison-Wesley (1984). [16] S. Lang, Linear Algebra, reprint of the third edition, Springer-Verlag, (1989). [17] S. Lang, Differential and Riemannian Manifolds, third edition, Springer• Verlag (1995). [18] S. Lang, Fundamentals of differential geometry, Springer-Verlag (1999). [19] H.B. Lawson & M.-L. Michelsohn, Spin Geometry, Princeton University Press, (1989). [20] C.R.F. Maunder, Algebraic Topology, Dover Publications (1996). [21] M.A. Na'imark, Normed Rings, translated by L.F. Boron, Wolters-Noord- hoff Publishing (1970). [22] G.K. Pedersen, Analysis Now, Springer-Verlag (1989). [23) LR. Porteous, Topological Geometry, Van Nostrand Reinhold (1969). [24] LR. Porteous, Clifford Algebras and the Classical Groups, Cambridge Uni• versity Press (1995). [25] H. Sato, Algebraic Topology: An Intuitive Approach, Amer. Math. Soc. Trans. of Mathematical Monographs 183 (1996). [26) J.-P. Serre, Complex Semisimple Lie Algebras, Springer-Verlag (1987). [27) S. Sternberg, Group Theory and Physics, Cambridge University Press (1994). [28) G. Strang, Linear Algebra and its Applications, Second edition, Academic Press [Harcourt Brace Jovanovich) (1980). [29) F.W. Warner, Foundations of Differentiable Manifolds and Lie Groups, Springer-Verlag (1983).

Additional entries added for the second printing

[30) J.C. Baez, The octonions, Bull. Amer. Math. Soc. 39 (2002), 145-205. [31) T. Brocker & T. tom Dieck, Representations of Compact Lie Groups, 3rd printing, Springer-Verlag (2003). [32) J.J. Duistermaat & J.A.C. Kolk, Lie Groups, Springer-Verlag (2000). [33) J. M. Lee, Introduction to Smooth Manifolds, Springer-Verlag (2003). Index

LPU-decomposition, 226 - Lorentz Clifford, 177 k-Lie algebra, 67 - norm, 111 k-line in k"+l, 217 - normed, 111 k-matrix group, 15 - real Clifford, 130 k-norm, 7, 34, 111 - semi-simple, 106 k-projective space, 217 - simple, 104 ±-grading, 136 alternating group, 153 n-manifold, 182 anti-homomorphism of skew rings, 120 atlas, 182 abelian Lie algebra, 69 automorphism action - canonical, 134 - adjoint, 49, 75, 86, 188 - inner, 124 - continuous, 38 - of algebras, 101 - group, 37 - outer, 124 - linear, 38 - smooth, 208 basis adjoint - dual, 290 - action, 49, 75, 86, 188 binary expansion, 131 - representation, 75 Borel subgroup, 18 affine bracket - geometry, 19 - Lie, 68 - group, 19 Bruhat decomposition, 226 - transformation, 19 algebra, 99 canonical - automorphism, 101 - automorphism, 134 - automorphisms - commutation relation, 205 -- group of, 122 Cartan - Clifford, 130 - decomposition, 280 - commutative, 100 - matrix, 297 - division, 100 Cauchy sequence, 10 - finite dimensional, 4, 99 central - group, 100 - idempotent, 104 - homomorphism, 101 - subfield, 102 - isomorphism, 101 centre, 69, 151 - Lie, 59, 67 - of a Lie algebra, 69

325 326 Matrix Groups: An Introduction to Lie Group Theory

- trivial, 70 covering centreless, 70 - finite, 274 character cross product, 68 - irreducible curve -- complex, 108 - differentiable, 55, 57, 183 characteristic polynomial, 51 - smooth, 183 chart, 182 circle group, 251 derivation property, 77 Clifford derivative, 74 - algebra, 130 derived subalgebra, 69 -- Lorentz, 177 determinant -- real, 130 - reduced, 116 -- universal property of, 132 diagonalisable, 51 - group, 139 diffeomorphism, 181 -- finite, 155 differentiable, 183 clopen, 17 - curve, 55, 57, 183 closed subset, 8 - homomorphism, 74 closure, 41 - map, 73 commutative algebra, 100 Differential Geometry, 181 commutator, 68 dimension, 72 - subalgebra, 69 direct commute, 68 - isometry, 23 compact, 12, 13 - product, 19, 71, 101 - topological space, 13 disc complement - open, 7 - orthogonal, 274 disjoint union, 21 complete, 10 division algebra, 100 - norm, 10 dual complex - basis, 290 - matrix group, 29 - linear, 289 - root - structure, 209 -- system, 301 complexification, 92 Dynkin diagram, 298 component - connected, 298 - path,241 - irreducible, 298 conjugate, 258 - hermitian, 28 elementary 2-group, 154 - quaternionic, 121 Euler characteristic, 256 conjugation exponential function, 47 - map, 187 extraspecial 2-group, 154 - quaternionic, 120 connected, 235 field - by a path, 241 - skew, 100 - Dynkin diagram, 298 finite - path,235 - Clifford group, 155 continuous - covering, 274 - group action, 38 - dimensional algebra, 4, 99 - homomorphism, 32 flow line, 63 - isomorphism form -- of matrix groups, 32 - skew symmetric, 229 - map, 7 - symplectic, 26, 229 coordinate function, 8 function coroot, 300 - coordinate, 8 - simple, 300 - exponential, 47 Index 327

- logarithm, 48 Hausdorff space, 14 fundamental root, 292 Heisenberg future pointing light cone, 158 - algebra, 205 - group, 203 Gaussian elimination, 226 - Lie algebra, 205 general linear group, 5, 35 hermitian generalised permutation matrix, 132 - conjugate, 28 generated by a subset, 144 - skew, 83 generator, 254 homeomorphism, 14 - topological, 254 homogeneous geometry - polynomial, 39 - affine, 19 - space, 214 - symplectic, 26 homomorphism Grassmannian, 222 - algebra, 101 group - continuous, 32 - k-matrix, 15 - differentiable, 74 - action, 37 - Lie, 74 -- continuous, 38 - of algebras, 101 -- smooth, 208 - of Lie algebras, 70 - affine, 19 - property, 57 - algebra, 100 hyperplane, 23 - alternating, 153 - reflection, 23, 291 - circle, 251 - Clifford, 139 ideal - finite Clifford, 155 - general linear, 5, 35 - Lie, 69 - Heisenberg, 203 idempotent, 104 - inner automorphism, 124 - central, 104 - isometry, 23 - indecomposable, 105 - Lie, 187 - orthogonal, 104 - Lorentz, 26 indecomposable idempotent, 105 - matrix, 15 indirect isometry, 23 - of algebra automorphisms, 122 inner - of path components, 242 - automorphism, 124 - one-parameter, 57 -- group, 124 - orthogonal, 21 - product, 227 - outer automorphism, 124 -- invariant, 269 - pinor, 143 -- Lorentz, 157 - projective -- non-degenerate, 227 -- linear, 125, 219 -- positive definite, 228 - quasi-symplectic, 233 -- real, 86 - quaternionic symplectic, 122 - products, related, 227 - special linear, 5 invariant, 87 - special orthogonal, 21 - inner product, 269 - special unitary, 28 irreducible - spinor, 144 - complex character, 108 - symmetric, 153 - Dynkin diagram, 298 - symplectic, 27 isometry, 21, 158 - topological, 12 - direct, 23 - unimodular, 5 - group, 23 - unitary, 28 - indirect, 23 - Weyl, 260, 292 - linear, 21 isomorphism Hamiltonian, 118 - continuous 328 Matrix Groups: An Introduction to Lie Group Theory

-- of matrix groups, 32 map - of algebras, 101 - conjugation, 187 - of Lie algebras, 70, 86 - differentiable, 73 - left multiplication, 187 Jacobi identity, 68 - right multiplication, 187 Jacobian matrix, 184 - smooth, 181, 182 Jordan matrix - block matrix, 52 - Cartan, 297 - form, 51, 53, 79 - diagonalisable, 51 - generalised permutation, 132 Kronecker symbol, 21 - group, 15 -- complex, 29 Lefschetz Fixed Point Theorem, 256 - Jacobian, 184 left - Jordan block, 52 - multiplication map, 187 - lower triangular, 171 - regular representation, 103 - orthogonal, 20 Lie - permutation, 153 - algebra, 59, 67 - subgroup, 15, 16 -- abelian, 69 - unimodular, 4 -- Heisenberg, 205 - unipotent, 18 -- homomorphism, 70 - upper triangular, 18, 171 -- isomorphism, 70, 86 maximal, 255 -- quotient, 70 - torus, 251 -- simple, 70 minimal polynomial, 52 - bracket, 67, 68 - group, 187 negative -- semi-simple, 276 - light cone, 158 -- simple, 274 - root, 293 - homomorphism, 74 - ideal,69 non-degenerate inner product, 227 -- proper, 69 non-trivial root, 279 - subalgebra, 69 norm, 5-7, 34, 111 - subgroup, 187 - complete, 10 light cone, 158 - operator, 6, 35 - future pointing, 158 - supremum, 6 - negative, 158 normaliser, 260 - past pointing, 158 normed algebra, 111 - positive, 158 null linear - root space, 279 - action, 38 - vector, 158 - dual, 289 - isometry, 21 one-parameter local trivialisation, 214 - group, 57 locally path connected, 235 - semigroup, 57 logarithm function, 48 - subgroup, 57 Lorentz open disc, 7 - Clifford algebra, 177 operator norm, 6, 35 - group, 26 orbit, 37 - inner product, 157 orthogonal, 90 lower triangular - complement, 274 - matrix, 171 - group, 21 -- special, 21 manifold, 182 - idempotent, 104 - smooth, 181 - matrix, 20 Index 329 outer - topology, 33, 212 - automorphism, 124 -- universal property of, 213 -- group, 124 real parabolic subgroup, 221 - Clifford algebra, 130 past pointing light cone, 158 - inner product, 86 path - part, 138 - component, 241 - quaternion, 119 - connected, 235 - symmetric bilinear form, 87 -- locally, 235 - symplectic group, 27 permutation matrix, 153 reduced - generalised, 132 - determinant, 116 Pfaffian, 156, 232 - echelon form, 226 phase portrait, 63 - root system, 292 pinor group, 143 reflection polynomial - hyperplane, 291 - homogeneous, 39 - in a hyperplane, 23 - minimal, 52 related positive - inner products, 227 - definite, 87 - symmetric -- inner product, 228 -- matrices, 227 - light cone, 158 representation, 38 - root, 293 - adjoint, 75 principal bundle, 214 - left regular, 103 product r~ght m~ltiplication - cross, 68 map, 187 rmg anti-homomorphism, 120 - direct, 19, 71, 101 root, 279, 292 - inner, 227 - fundamental, 292 - semi-direct, 19, 71 - topology, 11 - negative, 293 - vector, 68 - non-trivial, 279 projective - positive, 293 - linear group, 125, 219 - simple, 292, 293 - quaternionic symplectic group, 219 - space, 279 - space, 217 -- decomposition, 280 - sp:cial unitary group, 219 -- null,279 - umtary group, 219 - system, 292 proper Lie ideal, 69 -- dual, 301 pure quaternion, 119 -- reduced, 292 rotation, 23 quadratic form, 227 quasi-symplectic group, 233 semi-direct product quaternion - of groups, 19 - pure, 119 - of Lie algebras, 71 - real, 119 semi-simple quaternionic - algebra, 106 - conjugate, 121 - Lie group, 276 - conjugation, 120 semigroup - symplectic group, 122 - one-parameter, 57 -- projective, 219 separable, 182 quaternions simple, 70 - skew field of, 118 - algebra, 104 quotient - coroot, 300 - Lie algebra, 70 - Lie 330 Matrix Groups: An Introduction to Lie Group Theory

-- group, 274 - matrices, related, 227 - Lie algebra, 70 - real bilinear form, 87 - root, 292, 293 - skew, 26 skew symplectic - field, 100 - form, 26, 229 - hermitian, 83 - geometry, 26 - symmetric, 26 - group, 27, 122 -- form, 229 - group, quaternionic, 122 smooth - group, real, 27 - action, 208 - curve, 183 tangent space, 71, 184 - manifold, 181, 182 timelike vector, 158 - map, 181, 182 topological space - generator, 206, 254 - Hausdorff, 14 - group, 12 - homogeneous, 214 - space - projective, 217 -- compact, 13 spacelike vector, 158 topology special - product, 11 - linear group, 5 - quotient, 33, 212 - orthogonal group, 21 torus - unitary group, 28 - maximal, 251, 255 -- projective, 219 - of rank r, 252 sphere - standard, 252 - unit, 144 translation subgroup, 19 spinor group, 144 transpose, 20 stabiliser, 37 trivial centre, 70 standard torus, 252 strictly upper triangular, 81 unimodular, 4 subalgebra - group, 5 - commutator, 69 - matrix,4 - derived, 69 union - Lie, 69 - disjoint, 21 subfield unipotent - central, 102 - matrix, 18 subgroup - subgroup, 18 - Borel, 18 unit sphere, 144 - generated by a subset, 144 unitary - Lie, 187 - group, 28 - matrix, 15, 16 -- projective, 219 - one-parameter, 57 universal property, 132 - translation, 19 - of a Clifford algebra, 132 - unipotent, 18 - of the quotient topology, 213 - upper triangular, 18 upper triangular submanifold, 186 - matrix, 18, 171 subset - strictly, 81 - clopen, 17 - subgroup, 18 - closed,8 supremum norm, 6 vector product, 68 symmetric - group, 153 VVeylgroup, 260, 292