(a b) c = a (b c) e a e = e a = a. a a 1 = a 1 a = e. m+n

Μέγεθος: px

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Download "(a b) c = a (b c) e a e = e a = a. a a 1 = a 1 a = e. m+n"

Αλκιππη Δουμπιώτης
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3 Z 6 D 3

5 G = {a, b, c,... } G a, b G a b = c c (a b) c = a (b c) e a e = e a = a a a 1 = a 1 a = e Q = {0, ±1, ±2,..., ±n,... } m, n m+n m + 0 = m m + ( m) = 0

6 Z N = {a n }, n = 1, 2... N N Z N = {1, ω, ω 2,... ω N 1 } ω N = 1 ω = e 2πi/N a k a l = ω k ω l = ω k+l ω m k + l = m < N ω m Z N k + l = m > N m = nn + m m < N ω m = ω nn+m = ω m Z N e = ω N = 1 ω k ω N k = ω k a a 1 = a 1 a = e a k a l = a k+l ω N = 1 Z N a k a l = a l a k Q N = {1, 2, 3,..., N 1} (N) m, n (m + n)/n (N) v mn = m n = (m + n)( N) = m + n λn λ 0 < v mn < N (m n) p = (m+n λn) p = ((m+n λn)+p) µn = m+n+p νn (m (n p) = m (n+p κn) = ((m+n κn +p) ξn = m+n+p ρn N ν = ρ m G 0 < m < N N m G m + (N m) = 0 m, (N m) e = 0

7 (e, a) a 2 = e a 1 = a Z 2 (e, a 1, a 2 ) a 1 a 1 = e a 1 a 1 = a 2 a 1 a 1 = e (e, a 1 ) Z 2 a 1 a 1 = a 2 a 1 = a a 2 = a 1.a 1 = a 2 Z 3 e a 1 a 2 e e a 1 a 2 a a 1 a 2 e a 2 a 2 e a 1 Z 3 e a a 2 e e a a 2 a a a 2 e a 2 a 2 e a Z 3 (e, a 1, a 2, a 3 ) a 1 a 1 a 1 = a 2 a 1 a 1 = e a 1 a 1 = a 3 a 2 a 3 a 1 a 1 = a 2 Z 4 a 1 = a Z 4 a k = a k, k = 1, 2, 3 a 4 = e Z 4 e a 1 a 2 a 3 e e a 1 a 2 a 3 a 1 a 1 a 2 a 3 e a 2 a 2 a 3 e a 1 a 3 a 3 e a 1 a 2 Z 4 e a 1 a 2 a 3 e e a a 2 a 3 a a a 2 a 3 e a 2 a 2 a 3 e a a 3 a 3 e a a 2 Z 4 Z 2 (e, a 2 = a 2 ) ea 2 = a 2, a 2 a 2 = e Z 2 {e, a 2 = a 2 } Z 2 Z 4 a 1 a 1 = e (i)a 2 a 3 = e (ii)a 2 a 3 = a 1 (i) a 1 a 1 = e, a 2 a 3 = a 1

8 D 2 e a 1 a 2 a 3 e e a 1 a 2 a 3 a 1 a 1 e a 3 a 2 a 2 a 2 a 3 e a 1 a 3 a 3 a 2 a 1 e D 2 C ±π a i a j = a j a i D 2 Z 2 (e, a 1 ), (e, a 2 ), (e, a 3 ) a 1 a 2 = a 2 a 1 = a 3 D 2 Z 2 Quaternions D 2 = Z 2 Z 2 Laplace Hamilton quaternions i, j, k 1 quaternion q = a + b i + c j + d k 1 i 2 = j 2 = k 2 = i j k = 1 i j k = 1 i i (i j k) = i ( 1) i 2 j k = i j k = i i j k = 1 k (i j k) k = ( 1) k i j k 2 = k i j = k

9 i(i j) = i k i k = j j k = i j 2 k = j i j i = k (j i) i = k i k i = j i j = j i, j k = k j, i k = k i q 1 + q 2 = q 2 + q 1 q 1 + (q 2 + q 3 ) = (q 1 + q 2 ) + q 3 q 1 (q 2 q 3 ) = (q 1 q 2 )q 3 q 1 (q 2 + q 3 ) = q 1 q 2 + q 1 q 3 q + 0 = 0 1 q = q 1 = q quaternions q 1 q q q 1 q = qq 1 = 1 q = a b i c j d k q q = a 2 + b 2 + c 2 + d 2 q 1 = a b i c j d k a2 + b 2 + c 2 + d 2

10 {1, 1, i, i, j, j, k, k} {q 1, q 2, q 3, q 4, q 5, q 6, q 7 } e q 1 q 2 q 3 q 4 q 5 q 6 q 7 e e q 1 q 2 q 3 q 4 q 5 q 6 q 7 q 1 q 1 e q 3 q 2 q 5 q 4 q 7 q 6 q 2 q 2 q 3 q 1 e q 6 q 7 q 5 q 4 q 3 q 3 q 2 e q 1 q 7 q 6 q 4 q 5 q 4 q 4 q 5 q 7 q 6 q 1 e q 2 q 3 q 5 q 5 q 4 q 6 q 7 e q 1 q 3 q 2 q 6 q 6 q 7 q 4 q 5 q 3 q 2 q 1 e q 7 q 7 q 6 q 5 q 4 q 2 q 3 e q 1 G a G b a i b i : a i a j b i b j D 2 e, q1 e q 2, q 3 a 1 q 4, q 5 a 2 q 6, q 7 a 3 (G a ) = (e, q 1 )

11 H = {h i } G = {a j } G H a 1 i h j a i = h k H a i G Z 4 Z 2 (e, a 1 ) Z 4 N/n a 1, a 2 G a 2 a 1 a 1 1 a 2 a 1 a G a k a k = a 1 a k a a m = a k a l a m = a ka l = (a 1 a k a)(a 1 a l a) = a 1 (a k a l )a = a 1 a m a a G G a 1 Ga = G G H G a G ah = {ah 1, ah 2,..., ah n } S a = {a i }, S b = {b j } a i b j H G a G H G H G/H N/n h n h H

12 n h a i h m, a j h n (a i h m )(a j h n ) = a i (a j a 1 j h m )(a j h n ) = a i a j h mh n = a k h p a i H a j H = a k H H G/H H = (e, q 1 ) q 2 H, q 4 H, q 6 H a k C k C k : a k = a 1 i a k a i, a i G C k G C k a l C l a n a n = e a (a 1 k aa k) n = (a 1 aa k)(a 1 aa k)... (a 1 aa k)) = a 1 an a k = a 1 ea k = e k k k k k C k n k N/n k

13 (x 1, x 2, x 3 ) a 0 : (x 1, x 2, x 3 ) (x 1, x 2, x 3 ) a 1 : (x 1, x 2, x 3 ) (x 2, x 1, x 3 ) a 2 : (x 1, x 2, x 3 ) (x 3, x 2, x 1 ) a 3 : (x 1, x 2, x 3 ) (x 1, x 3, x 2 ) a 4 : (x 1, x 2, x 3 ) (x 3, x 1, x 2 ) a 5 : (x 1, x 2, x 3 ) (x 2, x 3, x 1 ) e ( ) , a1 ( ) , a2 ( ) a 3 ( ) , a4 ( ) , a5 ( ) ea i = a i, a 1 a 2 = a 5, a 2 a 1 = a 4, a 4 a 2 = a 3 a 4, a 5 a 1, a 2 a 3 a 3 = a 4 a 2 = a 2 a 1 a 2 S 3 e, a = a 1, b = a 2 S 3 = {e, a, b, bab, ba, ab} S 3 3! = 6 H = {e, ba, ab} H G (ba) 1 = ab C 1 = {e} a 1 i ea i = e a i S 3 a e 1 ae = a b 1 ab = bab (ab) 1 a(ab) = bab (ba) 1 a(ba) = b (bab) 1 a(bab) = b

14 (a, b, bab) a : (1, 2, 3) = (2, 1, 3) a 2 = e b : (1, 2, 3) = (3, 2, 1) b 2 = e (bab) 2 = (bab)(bab) = baeab = baab = beb = b 2 = e C 3 = {a, b, bab} (ba) 3 = (ab) 3 = e S 3 C S 3 : 1 : {e} 1 C 2 : {ab, ba} 3 C 3 : {a, b, bab} 2 S 3 H 1 : {e} H 2 : {e, bab}, {e, a}, {e, b} H 3 : {e, ba, ab} H 3 g G g 1 H 3 g = H 3 H 3 H 2 S n n S n ( ) n (p 1 p 2... p n ) p 1 p 2 p 3... p n

15 n n 1 n 3 n(n 1)(n 2) 2 1 = n! S n ( )( ) ( ) n p1 p 2 p 3... p n n = p 1 p 2 p 3... p n q 1 q 2 q 3... q n q 1 q 2 q 3... q n m < n n m S 4 ( ) = (12)(34) S 3, S 4 G a n, n = 1, 2,..., N H H G h i, i = 1, 2,... n < N a 1 a 1 G a 1 / H n a 1 h i a 1 h i / H a 1 h i = h j a 1 = h j h 1 i h j h 1 i H a 1 H {h i } {a 1 h i } a 2 = a 1 a 2 G a 2 / H {a 1 h i } {a 2 h i } H {a 1 h i } h k, h l a 1 h k = a 2 h l a 2 = a 1 h k h 1 l a 1 h m a 2 {a 1 h i } a, a,... a 1, a 2

16 {h i } {a 1 h i } {a 2 h i },... {a p 1 h i } G = H + a 1 H + a 2 H + + a p 1 H G N H n N G n N = pn p N Z 4 e, a, a 2, a 3 a 4 = e e, a 2 Z 2 = {e, a 2 } a Z 2 = {a, a 3 } Z 4 = Z 2 + a Z 2 = 4, n = 2 N/n = 2 a a, a 2,..., a k,... m a m = a m a a G e, a, a 2,..., a n 1 n a n N G N = 5 a 5 = e N = 7, 11 Z 6 D 3 Z 6 a 6 = e 1, a, a 2, a 3 a 4, a 5 Z 3 = (1, a 2 = a 2, a 4 = a 4 ), Z 2 = (1, a 3 = a 3 ) Z 6 a 2 a 3 = a 5 Z 6 Z 6 = Z 3 Z 2

17 A, B, C O OA, OB, OC 1, ω, ω 2 ω 3 = 1 π 3, 2 π 3 R 1, R 2, R 3 OA, OB, OC a, b a 3 = b 2 = e, bab 1 = a 1 n D n a n = b 2 = e, bab 1 = a 1 D 3

18 2 2 I = ( ) , A1 = ( ) , A2 = ( ) i 0 0 i, A3 = ( i 0 0 i A 4 = ( ) , A5 = ( ) , A6 = ( ) 0 i i 0, A7 = ( 0 i i 0 n GL(n, C) n g G g G = {g 1, g 2,... g N } g 1 1,... g N N g i ), ). g i : g j g k, g i g j = g k k = Γ(g i ) j Γ(g i ) Γ(g i ) jk = { 1 αν gi g j = g k 0 αλλιωσ

19 Z 3 e, ω, ω 2 ω 3 = ω ω : e ω ω : ω ω 2 ω : ω 2 ω 3 = 1 ω ω 2 N V i, j i = δ ij G N G D(g N N R g G g i i(g) = D ij (g) j D(g 1 ) = D(g) 1 ( ) D D(g) = (a) (g) 0 K(g) D (b) (g) D(g) = ( D (1) (g) 0 0 D ( ) (g) ) = D (1) D ( )

20 D(g) = D (1) D (2) D (k) V = V 1 V 2 V k V 1 a, b,... a, b = 1,..., d 1 < N a a(g) = D (1) ab b V 1 S d 1 N a = S ai i a 1 a 2... a d1 = S 11 (g)... S 1N S d S d1 M a i 1 i 2... i N a a(g) = D (1) (g) ab b b = S bi i a(g) = D (1) (g) ab S bj j a(g) = S ai i S ai i(g) = S ai D ij (g) j S ai D ij (g) = D (1) (g) ab S bj D(g) S a i D(g) D (1) (g)

21 a N N D(g) S 1 o S = 0 d 1 = N S N N a, i S D (1) (g)s = SD(g) D (1) (g) = SD(g)S 1 S = 0 S 0 R D(g) D(g)S = SD(g) g G S S = I S S i = s i i SD(g) i = D(g)S i S i(g) = s i D(g) i s i D(g) i = s i i(g) S i(g) = s i i(g) i(g) S S 2 o g G D(g)S = SD(g) S S = I

22 G N D a (g), D b (g) d a, d b D a (g) ik D b (g 1 ) jl = N δ ab δ ij δ kl d a g G U(g) U a (g) ik Ub (g) jl = N δ ab δ ij δ kl d a g G χ Da = D a (g) χ Da (g) χ Db (g) = Nδ ab g G χ Da (C i ) χ Da (C j ) = N δ Ci C n j i g G C i, C j n i C i N = n a=1 d 2 a d a

23 S 3 D(g) = 1, g G χ 1 (g) = 1 n C 1 = {e} n n C 1 n χ Da (C 1 ) = n m n n C i = C j = C 1 m n n 2 = m m m 3 + = N = 6 n 1 n N = 6 S 3 n 1 = 1 C 1 m n = 3 n m 1 + 4m 2 = 6, m 1 + m 2 = 3 m 1 = 2, m 2 = 1 1, 1 χ 1 (C i ), χ 1 (C i ), χ 2 (C i ) χ 1 (C i ) = 1 C 1 n n n χ 1 (C 1 ) = 1 χ 2 (C 1 ) = 2

24 a h i = 1 h i h 1 = 1, h 2 = 3, h 3 = 2 χ 1 (C 2 ) 3 = 1 χ 1 (C 2 ) = 1, ω, ω 2 χ 1 (C 3 ) 2 = 1 χ 1 (C 3 ) = 1, 1 D a = 1, D b = 1 g i χ 1 (g i )χ 1 (g i ) = χ 1 (C 2 ) + 3χ 1 (C 3 ) = 0 χ 1 (C 2 ) = 1, χ 1 (C 3 ) = 1 1, χ 2 (C 2 ) + 3χ 2 (C 3 ) = χ 2 (C 2 ) 3χ 2 (C 3 ) = 0 χ 2 (C 2 ) = 1, χ 2 (C 3 ) = 0 S 3 h χ 1 χ 1 χ 2 C C C S 3

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