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N = 4 M4 M4 H4 R1,4 H4 SO0(1, 4) R1,4 M4 = H4 H4 SO0(1, 4) H3 f(x, z) = n(z)n(x), x T4 n(z) x z Spin+(1, 3) / () = x, g | x Tn g Spin+(p, q) n = p + q R1 S3 S3 Z SO0(1, 4) j = 1/2 j > 1/2 SO0(1, 4) SO0(1, 3) SO(4) SO0(1, 4) SU(2) SU(1, 1) M = SO0(1, 4)/H H SO0(1, 4) H SO0(1, 4) Spin+(1, 4) Sp(1, 1) Spin+(1, 4) 2 2 SO0(1, 3) Spin+(1, 3) SL(2, C) Spin+(1, 3) 2 2 SO0(1, 4) SO0(1, 4) Spin+(1, 4) Sp(1, 1) Spin+(1, 3) SL(2, C) Sp(1, 1) SO0(1, 4) SO0(1, 4) () = x, q | x T5 q Spin+(1, 4) Sp(1, 1) C 1|U(g)|2 dg G G dg G U(g) G SO0(1, 4) U(g) SO0(1, 4) SO0(1, 4) SO(4) SO0(1, 3) N () = x, q | (l, 0) (0, l) l (1/2, 0) (0, 1/2) (1, 0) (0, 1) (l, 0)(0, l) (l1, l2)(l2, l1) SO0(1, 3) (l1, l2) (l2, l1) () = x, g | Spin+(p, q) Spin+(1, 3) SU(2) SU(2) P A A T A A P T A A A C Aut(C ) {1, P, T, P T } Aut(C ) C P T P T C A A C A A Aut(C ) C Ext(C ) Ext(C ) Ext(C ) Ext(C ) O(p, q) Rp,q O(p, q) O0(p, q) {1, P, T, P T } O(p, q) Pin(p, q) C p,q F = R O(p, q) Pin(p, q) C p,q {1, P, T, P T } C p,q {1, P, T, P T } {1, P, T, P T, C, CP, CT, CP T } Aut(C ) Ext(C ) CP T CP T CP T SO0(1, 3) SO(4) SO0(1, 4) Spin+(1, 3) SL(2, C) Spin(4) SU(2)SU(2) Spin+(1, 4) Sp(1, 1) N (l, 0) (0, l) SO0(1, 4) O0(1, 4) = SO0(1, 4) T5 M8 (l1, l2)(l2, l1) SO0(1, 4) O0(1, 4) P T C CP T CP T SO(n) SO0(1, n) Spin(n) Spin+(1, n) SO0(1, n) O0(1, n) = SO0(1, n) Tn SO0(1, 4) [Mk, Ml] = iklmMm, [Nk, Nl] = -iklmMm, [Pk, Pl] = iklmMm, [Mk, Nl] = iklmNm, [Mk, Pl] = iklmPm, [Mk, Nk] = [Mk, Pk] = [Mk, P0] = 0, [P0, Nk] = iPk, [P0, Pk] = iNk, [Pk, Nl] = iklP0, klm 1 k, l, m = 1, 2, 3 O0(1, 4) = SO0(1, 4) T5 SO0(1, 4) T5 R1,4 T5 T1 T5 = T1 T1 T1 T1 T1 T1 R+ O0(1, 4) O0(1, 4) [T, T] = 0, [T, J] = i(gT - gT), [J, J,] = i(gJ - gJ + gJ - gJ), J SO0(1, 4) T T5 O(1, 4) O(1, 4) = O0(1, 4) {1, P, T, P T } {1, P, T, P T } C SO0(1, 4) {1, P, T, P T, C, CP, CT, CP T } O(p, q) Pin(p, q) C p,q O(p, q) Spin+(p, q) Ca,b,c,d,e,f,g Pina,b,c,d,,e,f,g(p, q), Z2 Ca,b,c,d,e,f,g = {1, P, T, P T, C, CP, CT, CP T } CP T Ca,b,c,d,e,f,g SO0(1, 4) Spin+(1, 4) Sp(1, 1) a b a b Spin+(1, 4) H(2) : det = 1 = Sp(1, 1).

c d c d Spin+(1, 4) C + C + 1,4 1,4 C 1,4 R1,4 C + C 1,3 C 1,3 1,4 R1,3 C 1,3 C 1,3 C 1,1 C 0,2.

C 1,3 C 1,1 C 0,2 C 1,3 2 2 a+b1+c2+d12 1 = 2 = 1 = 2 = (12)2 = -1 123 124 2 2 2 2 = 1 = = = -1 1 2 1 2 3 4 1 i 2 j 12 k 4 4 4 4 4 4 AC = a0 + ai + aij + aijk +a1234 0 i i j i j k 1 2 3 1,i=1 i=1 j=1 i=1 j=1 k= C 1,AC = C 0 + C 1 1 + C 2 2 + C 3 12, 1,3 1,1 1,1 1,1 1, C i (i = 0, 1, 2, 3) 1, C 1, q Spin+(1, 4) g G G = KAK G K G A G SO0(1, 4) M = SO0(1, 3)/H M = SO(4)/H SO0(1, 3) SO(4) SO0(1, 3) SO(4) SO0(1, 4) SO0(1, 3) l0-l1+1 + + Tgq(, ) = ( + )l +l1-1( + ) q ;, + + l Pmn(cos ) l l l l e-i(m+n)Pmn(cos ) = e-ik Pmk(cos 1)Pkn(cos 2), k=-l 1 2 cos = cos 1 cos 2 - sin 1 sin 2 cos 2, sin 1 cos 2 + cos 1 sin 2 cos 2 + i sin 2 sin ei =, sin 1 2 2 1 2 2 i(+) cos cos ei - sin sin e-i 2 2 2 e =.

cos cos( - i) 2 = cos c = cos ch + i sin sh , sin ch - i cos sh ei = = 1, sin c i(+) cos ch + i sin sh 2 2 2 e = = 1.

c cos = = l l Zl (cos c) = Pmk(cos )Pl (ch ), mn kn k=-l l Pmk(cos ) = e-i(m+k)im-k (l - m + 1)(l + m + 1) (l - k + 1)(l + k + 1) cos2l tgm-k 2 min(l-k,l+k) i2j tg2j (j + 1)(l - m - j + 1)(l + k - j + 1)(m - k + j + 1) j=max(0,n-m) SU(2) Pl (ch ) = (l - n + 1)(l + n + 1)(l - k + 1)(l + k + 1) kn ch2l thn-k 2 min(l-n,l+k) th2s (s + 1)(l - n - s + 1)(n - k + s + 1)(l + k - s + 1) s=max(0,k-n) SU(1, 1) Zl (cos c) mn SU(2) SU(1, 1) Zl (cos c) mn Zl (cos c) mn m n m k, k n (l + m + 1)(l - n + 1) Zl (cos c) = im-n cos2l ch2l mn (l - m + 1)(l + n + 1) 2 l tgm-k thk-n 2 k=-l m - l, -k - l k - l, -n - l F1 - tg2 F1 th2, 2 m - k + 1 2 k - n + 1 Zl (cos c) m n m k, n k n m k m, n k mn n m k m, k n SO0(1, 3) - +i -1 a11z + a+i -1 2 2 Vaf(z) = (a12z + a22) (a12z + a22) f, a12z + a f(z) L2(Z) |f(z)|2dz < z = x + iy a T SO0(1, 3) S, Sl (a12z + a22) (a12z + a22) f(z) f(z) p(z, z) Sym(k,r) S, f(z) L2(Z) SO0(1, 3) 1 - +i,l0 - +i,l2 Mmn (g) = e-m( +i)-n(+i)Zmn = e-m( +i)-n(+i) l im-t (l0 - m + 1)(l0 + m + 1)(l0 - t + 1)(l0 + t + 1) t=-l cos2l tgm-t 2 min(l0-m,l0+t) i2j tg2j (j + 1)(l0 - m - j + 1)(l0 + t - j + 1)(m - t + j + 1) j=max(0,t-m) 1 ( + i - n)(1 + i + n)(1 + i - t)( + i + t) ch-1+2i thn-t 2 2 2 2 th2s, 1 (s + 1)( + i - n - s)( + i + t - s)(n - t + s + 1) 2 s=max(0,t-n) l0 = -k, -k + 1,..., k 2 - +i Mmn (g) SO0(1, 3) m t, t n (l0 + m + 1)(i - n + ) - +i,l2 Mmn (g) = im-ne-m( +i)-n(+i) (l0 - m + 1)(i + n + ) l cos2l ch-1+2i tgm-t tht-n 2 2 2 t=-l 1 m - l0, -t - l0 t - i +, -n - i + 2 F1 - tg2 F1 th2, 2 m - t + 1 2 t - n + 1 - +i,l Mmn (g) m t, n t t m, n t t m, t n SO(4) Spin(4) SU(2) SU(2) SO(4) SU(2) Spin+(p, q) C p,q Spin+(p, q) SO(4) SO0(1, 4) SO0(1, 4) SO(4) SO0(1, 3) SO0(1, 4) cos( + - i) = cos(e - i) = cos q 2 = cos q = cos e ch + i sin e sh , sin e ch - i cos e sh ei = = 1, sin q e e i(+) cos ch + i sin sh 2 2 2 e = = 1.

q cos = = Z (cos q) = Z (cos e)P (ch ), mn mk kn k=- Z (cos e) SO(4) mn Z (cos e) = Pmt(cos )Ptk(cos ).

mk t=- cos( + - i) = cos( + c) = cos q 2 = Z (cos q) = Pmk(cos )Z (cos c), mn kn k=- Z (cos c) = Pkt(cos )P (ch ) kn tn t=- SO0(1, 3) Z (cos q) mn SO(4) SO0(1, 3) Z (cos q) mn ( + m + 1)( - n + 1) Z (cos q) = cos2 cos2 ch2 mn ( - m + 1)( + n + 1) 2 2 m - , -t - im-k tgm-t tgt-k thk-n F1 - tg2 2 2 2 m - t + 1 t=- k=- t - , -k - k - , -n - F1 - tg2 F1 th2 t - k + 1 2 k - n + 1 m t, t k, k n Z (cos q) m t, k t, k n mn t m, k t, n k t m, t k, n k t m, k t, k n t m, t k, k n m t, t k, n k m t, k t, n k SO0(1, 4) g SL(2, C) q w = (az + b)(cz + d)- a b Sp(1, 1) Spin+(1, 3) SL(2, C) c d Spin+(1, 4) Sp(1, 1) SO0(1, 4) Sym(k,r) l0-l1+1 az + b az + b Tqq(z, z) = (cz + d)l +l1-1(cz + d) q ;, cz + d cz + d a, b, c, d H k = l0 + l1 - 1 r = l0 - l1 + 1 (l0, l1) SO0(1, 4) SO0(1, 4) l1 = - + i R l1 = - (n-1)+i SO0(1, n) SO0(1, 4) - +i,lMmn (q) = e-m( +i+k)-n(++i-j) (l0 + m + 1)(- + i - n) 0 cos2l cos2l ch-3+2i (l0 - m + 1)(- + i + n) 2 2 l0 l (l0 - k + 1)(- + i + k) im-k tgm-t tgt-k thk-n (l0 + k + 1)(- + i - k) 2 2 k=-l0 t=-l m - l0, -t - l0 t - l0, -k - l F1 - tg2 F1 - tg2 2 m - t + 1 2 t - k + 1 3 k + - i, -n + - i F1 2 th2, m t, t k, k n.

k - n + 1 - +i,l Mmn (q) m t, k t, k n t m, k t, n k t m, t k, n k t m, k t, k n t m, t k, k n m t, t k, n k m t, k t, n k O0(1, 4) = SO0(1, 4) T5 T() = x, q | R1, SO0(1, 4) M15 = R1,4 S10, H = 0;

M11 = R1,4 L6, H = {, , , };

M11 = R1,4 K6, H = {, , , };

M11 = R1,4 Sq, H = q ;

M9 = R1,4 H4, H = SO(4);

M9 = R1,4 Sc, H = {q, , };

M9 = R1,4 Se, H = {q, , };

M8 = R1,4 S3, H = {, , , , , , };

M8 = R1,4 H3, H = {SU(2), , , , };

M7 = R1,4 S2, H = {q, q, }.

M15 = R1,4 S O0(1, 4) S SO0(1, 4) M15 O0(1, 4) () = (x)(q) = (x0, x1, x2, x3, x4)(q1, q2, q3, q4, q5, q6, q7, q8, q9, q10), (x) T5 = T1 T1 T1 T1 T1 (q) M (q) mn - +i,lSO0(1, 4) Mmn (q) () T SO(2) f(q) SO0(1, 4) f() M15 = R1,4 S10 O0(1, 4) |f()|2d5xd10q < +, Sp(1,1) T SO0(1, 4) f() q f() = eipxe-i(m +nq)Z (cos q)d5x, mn mn =0 n=-T (-1)m-n(2 + 3)(2 + 3) = mn 32 q f()e-ipxe-i(m +nq)Z (cos q)d5xd10q.

mn Sp(1,1) T O0(1, 4) 1 | 2 := dg 1 | U(g) | 2.

gG U(g) SO0(1, 4) l l q l l Ml (q) = e-i(m +nq) Pmk(cos )Pkt(cos )Pl (ch ), mn tn k=-l t=-l l l Pmk(cos ) Pkt(cos ) SU(2) Pl (ch ) SU(1, 1) tn U(g) = AqKqAq SO0(1, 4) 1 | 2 := dd dd sin qe-m -n(+) 1 | P0KqP0 | 2, P0 SU(2) SU(2) P N 1 | 2 := dd dd sin qe-m -n(+)Pl (ch ) Pl (ch ).

m1,n1 mN,nN 1 (l - n + 1)(l + m + 1) Pl (ch ) = mn (m - n + 1) (l - m + 1)(l + n + 1) l + m + 1, m - l chm+n shm-n F1 - sh2.

2 2 m - n + 1 M4 = SO0(1, 4)/ SO(4) M H L H 1 | 2 = dd dd sh e-m -n(+)Pl (ch )Pl (ch ).

m1n1 m2n 1 | 2 = (m2 - n2 + 1) (l - m1 + 1)(l - n1 + 1)(l - n2 + 1)(l + m2 + 1) (l + m1 + 1)(l + n1 + 1)(l - m2 + 1)(l + n2 + 1) l-m1 m1+m2+n1+n2 m1+m2+s+p-n1-n2+ (-1)m +m2+n1+n2+p+k s=0 p=0 k=(l + m1 + s + 1)(m1 + m2 + n1 + n2)! (s + 1)(m1 - n1 + s + 1)(l - m1 - s + 1)(l + m2 - k + 2)! (m2 - n2 - k + 1)k (l + m2 - k + 1)k(m2 - l - k)k (m2 - n2 - k + 1)(-2l - 1) (-1)l+m-k+1 (m2 - l - k)(-n2 - l) l + m2 - k + 1, l + n2 + tm +s+p-n1-n2-l F1 + 2l + 2 t (m2 - n2 - k + 1)(2l + 1) 2-l-k (-1)m (l + m2 - k + 1)(-n2 - l) m2 - l - k, n2 - l tm +l+s+p-n1-n2+1 F1.

2 - 2l t F 1/t 1 | 2 tm +l+s+p-M-n1-n2+1 M 1 | M > m1 + l + s + p - n1 - n2 + M15 = R1,4 S10 O0(1, 4) TSO0(1, 4) (x) i + (x) = 0, xi (i = 0,..., 4) (x) T - (x) = 0.

i xi (q) k + (q) = 0, qk (k = 1,..., 10) (q) T - (q) = 0, k qk (q) 0 k (q) =, k =.

(q) k 0 = C0 I1, C1 I3,..., Cs I2s+1,...

1 2 0 = C I2, C I4,..., Cs I2s+1,...

Cs Cs s Cs cs l1,l2 l1,l 1 l1 = l1 l2 = l2 Cs 2 cs l1,l2 l1,l |l1 - l2| s l1 + l2, |l1 - l2| s l1 + l2.

2 M8 = R1,4 Sc Sc Sc Sc k 0 lm;l = fl (r)Ml (,, , , 0, 0), mn lmk k 0 l = fl (r)Ml (,, , , 0, 0), ;lm lk l0 l -l0 m n l0 l l -l l 0 0 Ml (,, , , 0, 0) mn 0 Ml (,, , , 0, 0) fl (r) lmk fl (r) lk R SO0(1, 3) (l, 0) (0, l) l SO0(1, 3) 1 - +i,l0 - +i,lk 2 lm;l = flmk (r)Mmn (,, , , 0, 0), 1 - -i,l - -i,lk 2 l;lm = flk (r)M (,, , , 0, 0), - < m, n < + - < , < + l l l0 ll0 + l1 - 1 l0 - l1 + l =, l =.

2 (1/2, 0) (0, 1/2) G+ 1, 0 0,.

2 P (1/2, 0) (0, 1/2) = (1, 2, 1, 2)T l + l 1n() = 1 (x)1n(g) = u1(p)e-ipxfl ( r)Ml (,, , , 0, 0), 1 1,,n 2 2 l + l 2n() = 2 (x)2n(g) = u2(p)e-ipxfl ( r)Ml (,, , , 0, 0), 1 1, -,n 2 2 l l 1() = 1 (x)1(g) = v1(p)eipxfl ( r)Ml (,, , , 0, 0), 1 1,-, 2 2 l l 2() = 2 (x)2(g) = v2(p)eipxfl ( r)Ml (,, , , 0, 0), 1 1,- -, 2 2 1/2 1 1/2 E + m 0 E + m u1(p) =, u2(p) = p-, pz 2m 2m E+m E+m p+ -pz E+m E+m ppz 1/2 E+m 1/2 E+m p+ -pz E + m E + m E+m E+m v1(p) =, v2(p) =, 2m 1 2m 0 p = px ipy fl ( r) = C1 cc rJl cc r + C2 cc rJ-l cc r, 1 1 , 2 C1 c fl ( r) = rJl+1 cc r 1 1 ,2 2 c C2 c - rJ-l-1 cc r, 2 c Jl cc r 1 3 l =,,,... ;

2 2 1 3 l =,,,... ;

2 2 1 2 Ml (,, , , 0, 0) = e ( +i)Zl (, ), l 1 1 2 Zl (, ) = cos2l ch2l i -k tg -k th-k 2 2 2 k=-l 1 - l + 1, 1 - l - k -l + 1, 1 - l - k F1 2 i2 tg2 F1 th2, 2 - k + 1 2 -k + 1 1 2 Ml (,, , , 0, 0) = e ( -i)Zl (, ), l 1 1 2 Zl (, ) = cos2l ch2l i -k tg -k th-k 2 2 2 k=-l 1 - l + 1, 1 - l - k -l + 1, 1 - l - k F1 2 i2 tg2 F1 th2 2 - k + 1 -k + l 1() = fl ( r) u1(p)eipxl,nMl (,, , , 0, 0)d4x, 1 1,,n 2 2 n=-lT l= l 2() = fl ( r) u2(p)eipxl,n Ml (,, , , 0, 0)d4, 1 1, -,n 2 2 n=-lT l= l 1() = fl ( r) v1(p)e-ipxl,Ml (,, , , 0, 0)d4x, 1 1,- , 2 2 l= =-lT l - 2() = fl ( r) v2(p)e-ipxl, Ml (,, , , 0, 0)d4x, 1 1,- -, 2 2 l= =-l T(-1)n(2l + 1)(2l + 1) l,n = 324fl ( a) 1, 2 F 1 ()e-ipxMl (,, , , 0, 0)d4xd4g, ,n Sc T(-1)(2l + 1)(2l + 1) l, = 324fl ( a) 1,2 1 ()eipxMl (,, , , 0, 0)d4xd4g.

, Sc T (1, 0)(0, 1) (1, 0) (0, 1) Sk SU(2) S, SO0(1, 3) S, (1, 0)(0, 1) G+ (1, 0) (0, 1) 1 (1, 0), (0, 1).

2 1 - +i,l0 - -i,l2 0,n () 0, () = (1, 2, 3, 1, 2, 3)T 1 - +i,l0 - +i,l2 1,n () = (k; x, t)1,n (g) = - 1 1 (k) - +i,l0 - +i,l2 2 2(2)3 exp[i(k x - t)]f1,1 (r)M1,n (,, , , 0, 0), (k) 1 - -i,l0 - -i,l 2 1, () = (k; x, t)1, (g) = - 1 1 (k) - -i,l - -i,l2 2 2(2)3 exp[-i(kx-t)]f1,1 (r)M1, (,, , , 0, 0).

(k) -k1k3 ik2|k| - 2 2 -k2k3 ik1|k| (k) = 2|k|2(k1 + k2) 2 k1 + k 1 l0 - +i,lf1,1 (r) = C r + l0 + i + + i + r, 2 2 2 1 3 l - -i,lf1,1 (r) = r + l - i + - i + r, 2 2 2 1 - +i,l0 - +i,l2 M1,n (,, , , 0, 0) = e( +i)Z1,n (, ), l - +i,l2 0 Z1,n (, ) = cos2l ch-1+2i,l i1-k tg1-k thn-k 2 2 2 k=-l 1 1 - l0 + 1, 1 - l0 - k n - i +, - i - k 2 F1 i2 tg2 F1 th2, 2 1 - k + 1 2 n - k + 1 1 - -i,l0 - -i,l2 M1, (,, , , 0, 0) = e( -i)Z1, (, ), l - -i,l0 2 0 Z1, (, ) = cos2l ch-1-2i,l i1-k tg1-k th-k 2 2 2 k=-l 1 1 - l0 + 1, 1 - l0 - k + i +, + i - k 2 F1 i2 tg2 F1 th2 2 1 - k + 1 - k + - 1() = 2(2)3 l 1 - +i,l0 (k) - +i,l2 f1,1 (r) eikxl,nM1,n (,, , , 0, 0)d4x, (k) l0=1 n=-l0T - 1 - -i,l2 1() = 2(2)3 f1,1 (r) l0= l (k) - -i,l e-ikxl,M1, (,, , , 0, 0)d4x, (k) =-l0T (-1)n(2l0 + 1)- +i,l1 = F1()e-ikxM1,n (,, , , 0, 0)d4xd4g, l0,n - +i,l324f1,1 (a)S Tc (-1)(2l0 + 1)- -i,l1 = 1()eikxM1, (,, , , 0, 0)d4xd4g.

l0, - -i,l 324f1,1 (a)S Tc SO0(1, 4) (1/2, 0) (0, 1/2) (1, 0) (0, 1) M LI D LI() = (()() ())(M ()).

k ()k L() = LD() + LM () + LI(), LD() LM () HI() = -LI() S + exp - i HI(x)d4x, S = T hc - T HI(x) M8 = R1,3 Sc R1, d4x Md8 = d4xd4g, d4g = sin cdddd.

M exp - i HI()d4xd4g.

S = T hc T4 S HI()dSc Sc f e e M (,, , , 0, 0)Mm (,, , , 0, 0)Mm (,, , , 0, 0) sin cdcdc.

le - +i l e Sc e im +e+mf I2 = (me + 1)(e + 1)(mf + 1) (le + me + 1)(l + e + 1)(mf + i + ) e (le - me + 1)(l - e + 1)(i - mf + ) e + c c e e sin +me+mf cos +me+mf 2 - le + me + 1, me - le l + e + 1, e - l c c e F1 e sin2 F1 sin2 2 e + 1 me + 1 1 i + mf +, mf - i + c 2 F1 sin2 sin cdc.

mf + 1 le-me e (le im +e+mf - me + 1)(l - e + 1)(mf + i + ) e I3 = (mf + 1) (le + me + 1)(l + e + 1)(i - mf + ) e 2 j1=l e-e (-1)j +j2(le + me + j1 + 1) e 2m +e+mf +j1+j2(j1 + 1)(j2 + 1)(me + j1 + 1)(e + j2 + 1) j2=me+e+mf q (l + e + j2 + 1) e (j1 + j2 + p)! (le - me - j1 + 1)(l - e - j2 + 1) e q=0 p=j1+j2+p (-1)q+p+k2p(me + e + mf )! p!(k - p)!(me + e + mf - q)!(mf + i + - k)! k=(mf - k + 1)k (mf + i - k + )k(mf - k + 1)k 1 (mf - k + 1)(-2i) f (-1)k-m -i- 1 (mf - i - k + )( - i) 2 1 mf + i - k +, i + 1 2 tj +j2+p-mf -i+ F1 + 2i + 1 t 1 (mf - k + 1)(2i) f (-1)k-m +i- 1 (mf + i - k + )( - i) 2 1 mf + i - k +, - i 1 2 tj +j2+p-mf +i+ F1, 1 - 2i t t = sin2 c I3 1/tM M I M > j1 + j2 + p - mf + i + S M H U(g) G H H g G : U(g) H | H C = ||||-2 | | U(g) |2d(g) < , gG d(g) G | H -| = C | U(g) d(g) U(g) | .

G G x = ax + b G 1 1 x - b daddb (x) = a(b), C ad a a 1 x - b a(b) = (x)ddx, ad a da C = |(ak)|2 < .

a () = x, g | G () = x, a | a G () = x, g | (x) = x | a = 1/ (h) = khk, k= h / / 1,.

2 3 1 1, 0 1,, 1 0,, 2 2 2 1 1,, 1.

2 (l1, l2) (l2, l1) kk 0 lm;l = fl l0 (r)Ml l0 (,, , , 0, 0), mn; lmk;lk 0 0 kk lm;l = fl l0 (r)Ml l (,, , , 0, 0), mn; lmk;lk l0 l -l0 m, n l0 l l -l , l 0 0 0 0 0 Ml l Ml l mn; mn; 0 0 0 fl l (r) fl l (r) lmk;lk lmk;lk CP T Cn F = C Ext(Cn) = {Id,,,,,,, } 1 P T P T C CP CT CP T 1 1 P T P T C CP CT CP T P P 1 P T T CP C CP T CT T T P T 1 P CT CP T C CP P T P T T P 1 CP T CT CP C C C CP CT CP T 1 P T P T CP CP C CP T CT P 1 P T T CT CT CP T C CP T P T 1 P CP T CP T CT CP C P T T P CP T Cn Ext(Cn) CP T Ext(Cn) {1, P, T, P T, C, CP, CT, CP T } Z2 Z2 Z2 P T P T C CP CT CP T A A A A A A A A A A A A A A {1, P, T, P T, C, CP, CT, CP T } P = T = (P T )2 = C2 = (CP )2 = (CT )2 = (CP T )2 = Z2 Z2 Z {Id,,,,,,, } (A ) = A (A ) = A A = A A = A = = = Id {1, P, T, P T, C, CP, CT, CP T } {Id,,,,,,, } Z2 Z2 Z2.

2 P = T =... = (CP T )2 = CP T Ext(Cn) Ext(Cn) Cn Id Id Id Id Id Id Id Id Id Id Cn A A A A A A A A A A A A A A A A Ext(C ) = {Id,,,,,,, } Ext(C ) CP T {1, P, T, P T, C, CP, CT, CP T } A A A A A A Ext(C ) A - A, A = WAW-1;

A - A, A = EATE-1;

A - A, A = CATC-1, C = EW;

A - A, A = A-1;

A - A, A = KAK-1, K = W;

A - A, A = S AT S-1, S = E;

A - A, A = F (A)T F-1, F = C.

Ext(Cn) C p,q K R Ext(Cn) Aut(Cn) Ext(Cn) C p,q K H p - q 4, 6 (mod 8) Ext(Cn) Aut(C p,q) Ext(Cn) Pin(n, C) O(n, C) Cn P T Pina,b,c(n, C) Pina,b,c(n, C) Pina,b,c(p, q) F = R Cn C p,q {1, P, T, P T } {Id,,, } Cn C p,q A A Cn {Id,,,,,,, } Z2 Z2 Z CP T O(n, C) (Spin+(n, C) Ca,b,c,d,e,f,g) Pina,b,c,d,e,f,g(n, C), ZCa,b,c,d,e,f,g = {1, P, T, P T, C, CP, CT, CP T } CP T Ca,b,c,d,e,f,g Ca,b,c,d,e,f,g Ext(C p,q) = Z Ca,b,c,d,e,f,g Z2 Z2 Z CP T R4,1 R1, (+, +, +, +, -) (-, -, -, -, +) C 4, R4,1 C 4,1 C4 C C 1,4 R1, C 1,4 C 1,3 C 1,3 C 1, CP T O(4, 1) C 4,1 CExt(C 4,1) = {I, 12345, 34, 125, 345, 12, 5, 1234}.

1 04 0 4 2 024 02 1 1 04 0 4 2 024 02 04 04 1 -4 -0 -024 -2 24 0 0 4 1 04 02 24 2 04 4 0 -04 -1 -24 -02 024 2 2 -024 -02 24 -1 04 0 -024 024 -2 24 -02 04 -1 4 -02 02 -24 -2 024 -0 4 1 -24 24 -02 024 -2 4 -0 04 - CP T R1, Ext(C 4,1) Z4 Z2.

O(4, 1) (Spin+(4, 1) C-,-,+,+,-,-,+) Pin-,-,+,+,-,-,+(4, 1), Z C-,-,+,+,-,-,+ Z4 Z2 Z CP T R4, O(1, 4) R1,4 CP T P T C {1, P, T, P T, C, CP, CT, CT, CP T } {1, 04, 0, 4, 2, 024, 02, 24}.

Z4 Z(+, +, -, -, -, +, -) CP T R1, C+,+,-,-,-,+,- Z4 Z2 Z2.

(Spin+(1, 4) Z4 Z2 Z2) Pin+,+,-,-,-,+,-(1, 4).

Z C 1,Ext(C 1,4) {I, W, E, C, , K, S, F} {1, 01234, 24, 013, 13, 024, 1234, 0}.

Ext(C 1,4) Z4 Z2 (+, -, -, -, -, +, +) (Spin+(1, 4) C+,-,-,-,-,+,+) Pin+,-,-,-,-,+,+(1, 4), Z C+,-,-,-,-,+,+ Z4 Z2 Z CP T R1, {1, P, T, P T, C, CP, CT, CP T } {I, W, E, C, , K, S, F} Z4 Z2.

C P T C 1, CP T Spin+(1, 3) Ca,b,c,d,e,f,g Pina,b,c,d,e,f,g(1, 3) Z SO0(1, 3) Ca,b,c,d,e,f,g = {1, P, T, P T, C, CP, CT, CP T } CP T Spin+(1, 3) P W T E P T C C CP K CT S CP T F {I, W, E, C, , K, S, F} Ext(Cn) Cn CP T (l, 0) (0, l) 0 0-l1+ Cl +l1-1,0 C0,l Spin+(1, 3) r S2k S C2 C2 C2 C2 C2 C2, k r k k r r (l0, l1) =, + 1 (-l0, l1) = -, + 1 CP T 2 2 2 (l, l ) (l, l ) 0 0-l1+1,l0+l1- Cl +l1-1,l0-l1+1 Cl Spin+(1, 3) S2k+r S2k+r C2 C2 C2 C2 C2 C k+r C2 C2 C2 C2 C2 C2, r+k k-r k+r (l0, l1) =, + 2 CP T (1, 0) (0, 1) (3/2, 0) (0, 3/2) (1, 1/2) (1/2, 1) Z C Z G+ Spin+(1, 3) F = C Cn n 0, 1 (mod 2) BWC = G(Cn, , ) Z BWC = G(Cn, , ) M = M0 M C F = R n 1 (mod 2) C C C n 0 (mod 2) BWC = G(Cn, , ) M = M0 M1 C G+ Spin+(1, 3) p - q 0 (mod 8) p - q 1 (mod 8) 0 0 Rl Rl Rl 0,2 0,2 p - q 2 (mod 8) 0 Rl Cl 7 p - q 7 (mod 8) Cl Hl p - q 3 (mod 8) 6 p - q 6 (mod 8) 0 Hl Hl Hl 4,6 4,p - q 4 (mod 8) p - q 5 (mod 8) BWR = G(C p,q, , ) M = M+ M- D G+ Spin+(1, 3) p+q l0 = C p,q p - q 1, 5 (mod 8) F = R C p,q BWR = G(C p,q, , ) Z BWR = G(C p,q, , ) M Spin+(p, q) Spin+(1, 4) Sp(1, 1) SO0(1, 4) Spin+(1, 3) SL(2, C) SO0(1, 3) Spin(4) SU(2) SU(2) SO(4) SO0(1, 4) SO0(1, 3) SO(4) H SO0(1, 3) H SO0(1, 3) SO(4) SO(4) Sp(1, 1) Spin+(1, 4) Sp(1, 1) SO0(1, 4) SO0(1, 4) SO0(1, 3) SO(4) SO0(1, 4) O0(1, 4) = SO0(1, 4) T SO0(1, 4) O0(1, 4) N M4 = H M15 = R1,4 S10 O0(1, 4) (l, 0) (0, l) M (l, 0)(0, l) SO0(1, 3) (1/2, 0) (0, 1/2) (1, 0) (0, 1) M (1/2, 0) (0, 1/2) (1, 0) (0, 1) S M M S M (l1, l2) (l2, l1) C A A Cn A A Ext(C ) = {Id,,,,,,, } C A A A A A A Ext(C ) Ext(C ) Ext(C ) CP T Aut(C ) Ext(C ) CP T CP T R4,1 R1,4 (+, +, +, +, -) (-, -, -, -, +) CP T CP T (1, 0)(0, 1) (3/2, 0) (0, 3/2) (1, 1/2) (1/2, 1) CP T C 1, R1, CP T o(4, 2) o(4, 2) O(4, 2) SU(2, 2) O(4, 2) O(5, 1) O(3, 3) Spin+(p, q) SO0(1, 3) SO(4) SO0(1, 4) Spin+(p, q) C + p,q M15 = R1,4 S10 O0(1, 4) M SO0(1, 4) (l1, l2) (l1, l2) SO0(1, 3) CP T / / / / / CP T / / / / / / / / / / / / / (1, 0)(0, 1) / / / (1/2, 0) (0, 1/2) (1, 0) (0, 1) / (1/2, 0) (0, 1/2) / / / / / / / / / / / / / / / / /

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