1、? ØC q x ±, y ±q =0, 1, q n =1þ g Óéu f (x, y =a k,l x k y l , Eed k (f (x, y =a k,l (x k (y (l l d x, y 3e ”û½" 2k 7òC q ¥ü N ü §K ·:x xh y l ,y xi y j ,h, k, i, j Z,C d yx =qxy k (yx =(qxy O (yx =(y (x =qhj x i +h y j +k (q

2、xy =q(xy =q(x (y =qqki x h +i y k +jk q ki +1=q hj =hj ki =1e Ph iA =, (, (C , C ,k jh, k, i, j Z K k detA =1, Ó 7L ±$K §=éu f (x, y , g (x, y C q k (f (x, y g (x, y =(f (x, y (g (x, y ¤á§= = f (x, y , g (x, y üª¤á§ ?¿üªxn y

3、m , 1x n 1y m 1O (1x n y m x n 1ym 1=1n +n 1m +m 1q n 1m +(n +n 1(n +n 11 hk +(m +m 1(m +m 11 ij11q ki (n +n 1(m +m 1 x h (n +n 1+i (m +m 1 y k (n +n 1+j (m +m 1(xn y m =(x n n (y m m =n m q n (n 1 hk +m (m 1 ij +nkim x hn +im y kn +jm(1x n 1y m 1 =1(x n 1 n 1(y m 1 m 1=1n 1m 1q n 1(n 11 hk +m 1(m 1

4、1 ij +n 1kim 1x hn 1+im 1y kn 1+jm 11111 = hj ki =1¤á"e ¡? Øéu xn y m C q 3±½Ó e ´Äk ”§Ù ´Äo " · X J 3K 7üª§ 1x n 1y m 1§(1x n 1y m 1 =1(x n 1 n 1(y m 1 m 1=1n 1m 1q n 1(n 11 hk +m 1(m 11 ij +n 1

5、kim 1x hn 1+im 1y kn 1+jm 1=xn y m11=éA §|£C, m, n Z ; q ; h, k, i, j ; hj ki =1®¤1n 1m 11 n 1(n 11 hk +m 1(m 11 ij +n 1kim 1=q 1hn 1+m 1i =n n k +m j =m 113n 1, m 1Z, C S ´Äk § ´Ä"d ®hj ki =1´ d § m, n Z ½Ñk §Ï

6、;d xn y m C q 3±½Ó e ´k ”" C q g Ó 7g Ó "11±1Aut C q x ±1, x 2ÏAutC q k d x 1, y 13e ”û½" y ²AutC q òC q ¥ü N ü §:C q C q ;x xh y k , y xi y j,C , h, k, i, j Z ·PA =h ki j ,(, (C , C ,h, k, i, j

7、Z l d A, (, û½, d yx =qxy k detA =1§k A sl (2, Z . ·8Üsl (2, Z ×(C , C , ½ÂN :AutC q sl (2, Z ×(C , C , (A, (, d m © ©ÛV "e AutC q k:C q C q ;x x h y k , y x i y j, C , h , k , i , j Z K ( =(A , (, §Ù¥hA =k O ki,j(, (C ,

8、 C ( =(AA , (h k Q 1(A, A , i j Q 2(A, A Ù¥Q 1(A, A Q 2(A, A =q q1 1h (h 1 hk +k (k 1 ij +kh ik 1 1i (i 1 hk +j (j 1 ij +ki ij·½Âsl (2, Z ×(C , C ¥ ¦X e µ(A, (, (A , (, =(AA , (h k Q 1(A, A , i j Q 2(A, A w , ½ÂÜn ; X ·²sl (2, Z &

9、#215;(C , C U d ¦¤+, ´µ45¤á§k ü (E, (1, 1 3§e ¡y ² ´_ 3±9(ÜƤá"éu (A, (, (sl (2, Z ×(C ×C , (A , (, (sl (2, Z ×(C ×C ¦(A, (, (A , (, =(E, (1, 1 oAA , h k Q 1(A, A , i j Q 2(A, A =(

10、E, (1, 12=AA=E i j Q (A, A 2h k Q 1(A, A =1=1§ h =j k =ki =ij =h 1h k =Q (A, A 1 1=i j Q 2(A, A u ´ù (A , (, (sl (2, Z ×(C ×C 3¶, O (A , (, (A, (, =(E, (1, 1 § 8Ü¥ ? Ñk _ "y 3· (ÜƧ(A, (, , (A , (, , (B, (1, 2 (sl (2, Z ×

11、;(C ×C Ph 1i 1h i h iB =A =A =k j k 1j 1k jAA =hh +ikhi +ij ki +jjkh +jkA B =h h 1+i k 1k h 1+j k 1h i 1+i j 1k i 1+j j 1-M=(A, (, A , (, B, (1, 2=AA , h k Q 1(A, A , i j Q 2(A, A B, (1, 2=AA B,1(Q 1(A, A (Q 2(A, A k 1Q 1(AA , B ,=s 1hkh 1ij1(h k Q 1(A, A i 1(i j Q 2(A, A j 1Q 2(AA , B =t 1N=A,

12、 (, (A , (, B, (1, 2 A, (, AA B,A B, 1( h 1( k 1Q 1(A , B , 1( i 1( j 1Q 2(A , B =1( h 1( k 1Q 1(A , B h h 1+i=s 2k 1kh 1+j k 1Q 1(A, A B ,31( i 1( j 1Q 2(A , B h i 1+i =t 2j 1ki 1+j j 1Q 2(A, A B M =N s 1=s 2, t 1=t 2s 1=1(h k Q 1(A, A h 1(i j Q 2(A, A k 1Q 1(AA , B =1( h 1h =1( h 1h s 2=1( h 1kh 1

13、k 1i 1Q h 1(A, A ( k 1jk 11Q k 2(A, A Q 1(AA , B h 1+i k 1( k 1k ( h 1+j k 1k 11Q h 1(A, A Q 2(A, A Q 1(AA , B h 1h h 1+i k 1k 1k h 1+j k 1Q 1(A, A B Q 1(A , B s 1=s 2Q 1(A, A Q 2(A, A Q 1(AA , B Q 1(A, A B Q 1(A , B k 11Q h 1(A, A Q 2(A, A Q 1(AA , B =Q 1(A, A B Q 1(A , B =q h =q i111(h 1 hk +k (k

14、1 ij +kh ik1(i 1 hk +1j (j 1 ij +ki ij=q h 1(h 11(hh =q =q+ik (kh +jk +1k 1(k 11(hi +ij (ki +jj +(kh +jk (hi +ij h 1k 111(h h 1+i k 1(h h 1+i k 11 hk +(k h 1+j k 1(k h 1+j k 11 ij +ki (h h 1+i k 1(k h 1+j k 1 11h 1(h 11 h k +k 1(k 11 i j +k i h 1k 1k 11Q h 1(A, A Q 2(A, A Q 1(AA , B ê=11h 1h (h 1 hk +h 1k (k 1 ij +h 1kih k +221k i (i 11221 hk +1k 1j (j 1 ij +k 1kii j +h 1(h 11(hk (h +hjh k +kik h +ijk 22+1k 1(k 11(hk (i +hji j +kii j +ijj +h 1k 1(hki h +kih j +hjk i +ijk j 1=1(h h 1+i k 1(h h 1+i k 11 hk +(k h 1+j k 1(k h 1+j k 11 ij +11(h h 1+i k 1(k h 1+j k 1 ki +h 1(