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Контракции квантовых симплектических групп.

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???????? ???? ???????? ?????? ??? ???
?????? 4(8). ?????????, 2011.
ФИЗИКО-МАТЕМАТИЧЕСКИЕ НАУКИ
??? 517.95
КОНТРАКЦИИ КВАНТОВЫХ СИМПЛЕКТИЧЕСКИХ ГРУПП
Н.А. ГРОМОВ, В.В. КУРАТОВ
Отдел математики Коми НЦ УрО РАН, г.Сыктывкар
gromov@dm.komisc.ru, kuratov@dm.komisc.ru
Изучаются контракции квантовых симплектических групп Spq (n), рассматриваемых как алгебры Хопфа некоммутативных функций над алгебрами с нильпотентными образующими, а также контракции ассоциированных с ними квантовых симплектических пространств Sp2n
q . Развита общая теория для произвольного n, включающая разные сочетания схемы контракций Кэли-Клейна и
структуры алгебры Хопфа, которые описываются с помощью перестановок nго порядка. Общая теория подробно иллюстрируется при n = 2.
Ключевые слова: алгебра Хопфа, квантовая симплектическая группа, контракция
N.A. GROMOV, V.V. KURATOV. CONTRACTIONS OF QUANTUM SYMPLECTIC GROUPS
Contractions of the quantum symplectic groups Spq (n) and of the quantum symplectic spaces Sp2n
q are investigated. Contracted groups are regarded as Hopf algebras
of the noncommutative functions over the algebras with nilpotent generators. The
different combinations of the Cayley-Klein scheme of group contractions and the
noncommutative quantum structure are studed with the help of permutations of
n-th order in the framework of general theory for arbitrary n. The case of n = 2
is considered in detail.
Key words: Hopf algebra, quantum symplectic group, contraction
?????. ??????? ???????? ??????????????? ????????? ?????????? ????????????? ????? ?? ? ??????
?? ???????????? ??????????? ??????? ??? ? ?????
?????? ?? ?????????? ? ??????, ??? ? ?????????? ??????????????, ????????? ????????? ????????? ??????????? (??????????????) ????????????? ????? ??
? ?????? ?? ??????????. ????????????? ?????? ??
????? ???????? ?? ??????? ????? ??????? ?????????? [4], ??????? ??????? ?? ???????? ? ?????????
?????? ?????? ??? ?????????? ?????????? ????? ???????, ????? ??? ?????????? ???? ?????????? ? ???? ????????? ??????? ???????????, ? ??????????
?? ???? ????????? ?????????.
?????????????? ?????? [5] ? ???????? ????????????? ????? ?? ??????????? ? ????????????
????? ??? ?????????, ???????????? ?????????????? ?????????????? ???????????. ???????? ?????
????? ???????????? ????? ?????????? ???????, ????????? ???????? ??????? ????????? ???????? ?
???????? ? ?????????????? ???????????. ???? ??
?????? ???????? ? ? ?????? ????????? ????? [6]? [11].
? ??????? ?? [7,9] ? ????????? ?????? ?? ????????????? ?????? ??????????? ????????? ????????? (???
????????? ??????? ?????) ? ????? ?????????? ????-?????? ? ?????? ?????? ???????: ????????? ??????????????? ?????? ??? ???????????????? ?????????? ???????????????? ????????????.
????????
?????? ??????????? ??????? ????????????
????? ? ?????? ?? ??????????? ????? ? ?????????????, ????????? ? ??????????????? ? ?????????????? ???? ?????????????? ? ?????? [1], ??????????
????? ??????????? ? ??????? ??????? ???????? ????????? ? ??????? ?????????????? ???????????. ? ?????????????? ????? ?????? ????????? ?????? ?? ?
????????? ??????? ?? ???????????? ????? ???????
?????, ? ?? ????????, ?????? ????? ???????, ??????? ? ???, ??? ??? ???? ?????? ????????????? ??????? ??????????????? ? ???????????? ????????????????? ?????? ?????. ? 90-? ??. ???????? ????????
????????? ?????? ? ??????? ????????? ??????? ???????????? ? ?????????? ????????????? ??? ????????????, ??? ? ????????. ? ???? ??????????? ??????? ? ???????????? ? ??????????? ???????? ????????????????? ? ???????? ????, ? ?????????, ? ????????? ????????????-??????? [2, 3].
?????? ? ???, ? ????????????? ?????? ???????????? ?? ?????? ???????, ?? ? ????????????? ?????? ? ??????? ??. ? ?????????, ?????? ?????????????? ??????????????? ????????????-??????? ??????????? ?????? ??????????????? ???? ?????????????
??????, ?????????????? ????? ?????????? ???????????? ???????? ????????? ?????????? ? ?????? ??4
???????? ???? ???????? ?????? ??? ???. ?????? 4(8). ?????????, 2011
???????? ? N = 2n-??????? ?????????? ???????????????? ???????????? SpN
q (C), ???????????? ??????????? x?1 , . . . , x?2n ? ??????????????? ?????????????
R?q (x? ? x?) = q(x? ? x?),
(10)
1. ????????? ??????????????? ??????
? ????????????
????????? ??????????????? ?????? Spq (n)
???????????? [1] ?????? ?????????? ????????? ????????????? ??????. ??? ????????????? ? ????????
Rq ????
Rq
=
q
N
?
eii ? eii +
i=1
i?=i?
+q
?1
N
?
eii ? ekk +
x?i? x?i = x?i x?i? + (q 2 ? 1)
i,k=1
i?=k,k?
N
?
N
?
1 ? i < k ? 2n, i =
? k? ,
x?i x?k = qx?k x?i ,
ei? i? ? eii + ?
i=1
i?=i?
??
??? R?q = P Rq , P (a ? b) = b ? a. ? ????? ????
N
?
?
i?
?1
q ?i? ??l ?i? ?l x?l x?l? ,
l=1
?
eik ? eki ?
1 ? i < i ? 2n.
i,k=1
(11)
? ??????? SpN
q (C) ??????????? ?????????
i>k
q ?i ??k ?i ?k eik ? ei? k? ,
(1)
x?t C x? =
N
?
i,k=1
q ??k ?k x?k x?k? = 0,
(12)
k=1
i>k
??? N = 2n, ? = q ? q ?1 , (?1 , . . . , ?2n ) = (n, n ?
1, . . . , 1, ?1, . . . , ?n + 1, ?n), ?i = 1 ??? i = 1, . . . , n ?
?i = ?1 ??? i = n + 1, . . . , N , i? = N + 1 ? i. ? ??????? ???????? ??????? Rq ???????????? ?????????????? ???????????
? ?????? ???????? ????????? ??????????? ? (11)
????? ?????????? ? ????
(2)
????????? ??????????????? ???????????? Sp2n
q (C)
???????????? ? ????????? ?????? Spq (n) ??? ??????
??????? ??????? ?????????? x?k = tk1 , k = 1, . . . , n.
?? ????????? ?? ?????? ??????? R ???????? ? ????????? ?????? Spq (n) ?? ????????? ???????????????
???????????? Sp2n
q (C)
Rq T1 T2 = T2 T1 Rq ,
[x?i , x?i? ] = ?q
T =
tik
t i? k
tik?
t i? k ?
)
,
(3)
i, k = 1, . . . , n
????????? ?????? Spq (n), ??? T1 = T ? I, T2 = I ? T,
? I ? ????????? 2n Ч 2n ???????. ? ??????? Spq (n)
????? ????? ?????????????? ??????????? ????
T t C ?1 T = C ?1 ,
T CT t = C,
q i?k x?k? x?k ,
?
?(x?) = T ?x?,
?(x?i ) =
n
?
tik ? x?k , i = 1, . . . , n
????????? ?????????? ?????
??? C = C0 q ? , ?????? ? = diag(?1 , . . . , ?N ), (C0 )ij =
?i ?i? j , i, j = 1, . . . , N ? C 2 = ?I.
??????? ??????? ?? ????????? ?????? Spq (n) ?
???????? ???????? ????? ? ???????????? ?
? y? =
x?t ?C
?
?T = T ?T,
?tij =
tik ? tki ,
=
(5)
2. ?????? ?????????? ????????? ?????????
? ????? ?????????? ????-??????
?????? ????????? ?????????? ? ????? ?????????? ????-?????? ??????????? ??????? ?? ?????? ?????? ? ??? ??? ?????????? ? ???????? ????????? ??????? ????????? ?????????? ????????? ????????????. ??? ????????? ?????????? ???????????
? ??? ??????, ??? ????????????? ????????? ?? ?????? ?????????????? ???????? ????????????. ??????
? ??? ? ????????? ?????????? ? ?????????? ???????? ????????? ??????????????. ??? ??????????
????????????? ????????? ?????????? ??????????
????????????????, ?????? ?????????????? ??????????? ????? ??????????? ??????????? ???????
?? ?? ??????. ????? ?????????? ????-?????? ???????? ??????????? ?? ????????? ?????????? ????????: xk ? (1, k)xk , ?? ??? ??????? ?? ?????? ??????????. ??? ??????????? ????????? ????? ???? ?????????? ? ????? ??????? ? ????????? ?????? ???
(6)
?(T ) = I, ?(tij ) = ?ij
? ????????? S
S(ti? k ) = ?q i
?
?k+1
i, j = 1, . . . , n,
?
?1
tki? ,
tk? i , S(ti? k? ) = q k?i tki = q i
?
?k?
tki ,
(7)
?????????? ?????????
S 2 (T ) = (CC t )T (CC t )?1 .
(8)
????? ?????????? f (t) = t ? q ? ???????????
????????? ???????????
f (R?)(x ? x) = 0
)
q ?(n+1?k) x?k ? y?k? ? q (n+1?k) x?k? ? y?k , (15)
? y?) = 1 ? x?t ?C
? y? , ??? m : Spq (n) ?
?.?. m(? ? ?)(x?t ?C
Spq (n) ? Spq (n) ???? ??????????? ?????????.
?????????? ?
S(tik ) = q i?k tk? i? , S(tik? ) = ?q i?k
n (
?
k=1
k=1
S(T ) = CT t C ?1 ,
(14)
k=1
(4)
2n
?
(13)
i = 1, . . . , n.
k=1
??????????
(
i
?
(9)
5
???????? ???? ???????? ?????? ??? ???. ?????? 4(8). ?????????, 2011
????????? ???????????? ? ??????? ????????? [6].
??? ????? ?????????? ???????? ?????? ???? ?? ????????, ??????, ????????? ?????????? ? ??????? ???????????? ? ? S(n) ???????? ?????????? ?????????? ?????????? ???????????? ? ??????????, ???????? ?? ?????????????? ???????????, ? ????? ???????????? ???????? ??????? ?????????? ?????????
?????? ? RT T -??????????. ????? ?????????? ?????????? ??? ???? ????????? ??????????.
? ?????? ??????????????? ????? ? ???????????
??? ???????? ???????? ????????? ???????. ??????? ? ?????????? (10) ?????????? x?t = (x1 , . . . , xn ,
xn? , . . . , x1? ) ?? xt = (x?1 , . . . , x?n , x?n? , . . . , x?1? ) ? ????????? ????????? ??????????????? ???????????? ????-?????? Sp2n
v (j; ?) ?????????
??? i, k = 1, . . . , n. ?????????????? ??????????? ?????????? T (j; ?) ????????? ?? ??????? ?????????
(?????? (2))
Rv T1 (j; ?)T2 (j; ?) = T2 (j; ?)T1 (j; ?)Rv ,
? ?????????????? ??????????? (4) ????????? ???
T (j; ?)C(j)T t (j; ?) = C(j),
T t (j; ?)C ?1 (j)T (j; ?) = C ?1 (j),
? (j; ?),
?T (j; ?) = T (j; ?)?T
(?k , ?k ) ? 1.
(17)
3. ????????? ??????????????? ?????? Spv (2; j; ?)
? ????????? ???????????? Sp4v (j; ?)
l=min(?i ,?k )
?????????????? ?????????? ??????? ?????????
??????????????? ????????? ?????????? q = ez
???? z = Jv , ? ??????? ??????????? ?????????
J ????????? ?? ??????? ?????????????? ???? ?????????????? ??????????? ?????????? ????????????
??? n = 2 ????????? ??????????????? ???????????? Sp4q ??????????? ??????????? x1 ,x2 ,x2? ,x1? .
????? i = 1, 2, i? = 5 ? i, i? = 2? , 1? , ??? 2? = 3, 1? = 4.
??? ???????? ?? Sp4q ? Sp4v (j; ?) ??????????? ?????????????? (16), ? ??????: x?1 ? (1, ?1 )x?1 , x?2 ?
(1, ?2 )x?2 , x?2? ? (1, ?2 )x?2? , x?1? ? (1, ?1 )x?1? . ? ?????????? ?????????????? ??????????? ????????? ?
????
Sp2n
v (j; ?)
1 ? i < k ? 2n, i ?= k? ,
x?i x?k = eJv x?k x?i ,
i
2eJv sinh Jv ? J(i?k)v
e
(1, ?k )2 x?k? x?k ,
(1, ?i )2
[x?i , x?i? ] =
(23)
?? ??????? ????? ?????????????? ????????? ????????? ??????????????? ????? Spv (n; j; ?) ? ??????????????? ? ???? ????????? ???????????????
??????????? Sp2n
v (j; ?). ??????????????? ?????????? ??????????? ??? n = 2 ? ?????? ????????????? ??????, ??????????? ???? ?????????????? ????????.
max(?i ,?k )?1
jl ,
?(T (j)) = I,
S(T (j; ?)) = C(j)T t (j; ?)C ?1 (j).
??? ?????????? x??k ??????????? ??????? ??????????? x?k ? Rn (j), ? ?????????? x??k? ? R?n (j)
? ??????? ??????????? ? ????????? ????????????
Rn (j) ? R?n (j). ???????????? ?????????????? ?????????? ????????? ????????
?
(22)
??? Rv = Rq (q ? eJv ), C(j) = C(q ? eJv ). ????????? ??????? ????? ? ??????? ??????? ?? ?????????
?????? Spv (n; j; ?) ???????? ?????????????
x??k ? (1, ?k )x?k , x??k? ? (1, ?k )x?k? , k = 1, . . . , n, (16)
(?i , ?k ) =
(21)
x?1 x?2 = qx?2 x?1 ,
x?1 x?1? = qx?1? x?1 ,
x?2 x?1? = qx?1? x?2 ,
x?1? x?1? = qx?1? x?1? ,
k=1
(18)
i = 1, . . . , n
x?1 x?1? = q 2 x?1? x?1 ,
? ?????????????? ??????????? (21) ????????? ?????? Spv (n; j; ?). ? ?????? ????????????? ???????????? ?0 = (1, 2, . . . , n) ??????????? ????????? J =
(1, n)2 .
?????????????? ?????????? ????????? ??????????????? ?????? ?????? ?? ??????? ??????????
?????????? ????? ???????????? Sp2n
v (j; ?)
x?2 x?2? ? q 2 x?2? x?2 =
n
?
(
(1, ?k )2 e?J(n+1?k)v x?k ? y?k? ?
k=1
)
?eJ(n+1?k)v x?k? ? y?k .
[t?1 ?2 , t?2? ?1 ] = [t?1 ?2? , t?2? ?1 ] = [t?1 ?2 , t?2 ?1 ] =
(19)
= [t?2? ?1? , t?1? ?2? ] = [t?2? ?2 , t?1? ?1 ] = [t?2 ?1 , t?1 ?2? ] =
? ?????????? ??????? ??????? ?????????? ????????? ?????? Spv (n; j; ?) ????
(
)
T (j; ?) =
(
=
Tik
Ti? k
(?i , ?k )t?i ?k
(?i , ?k )t?i? ?k
Tik?
T i? k ?
= [t?2 ?1? , t?1? ?2? ] = [t?2? ?1? , t?1? ?2 ] = [t?2 ?1? , t?1? ?2 ] =
=
(?i , ?k )t?i ?k?
(?i , ?k )t?i? ?k?
(24)
??? q = eJv , ? = q ? q ?1 = 2 sinh Jv, J = j 2 . ???
????????? ?? ?????????? ??????????? j ?????? j1 .
?????????? ????????? ???????????????
?????? Spv (2; j; ?) ????????????? ?? ??????? (20),
??? i, k = 1, 2. ?????????????? ??????????? ?????????? ????????? ???????? ??????? ????????? (21)
? ?????????????? ?? ????????? ???? ?????:
1. ????????????? [a, b] = 0,
?
xt (j; ?)?C(j)y(j;
?) =
=
(1, ?1 )2
?x?1 x?1? ,
(1, ?2 )2
= [t?2 ?2 , t?1? ?1 ] = [t?2 ?2? , t?1? ?1 ] = [t?2? ?2? , t?1? ?1 ] =
)
= [t?2 ?2? , t?1 ?1? ] = [t?2? ?2 , t?1 ?1? ] = [t?2 ?2 , t?1 ?1? ] =
,
(20)
= [t?1? ?1 , t?1 ?1? ] = [t?2? ?2? , t?1 ?1? ] = 0,
6
(25.a)
???????? ???? ???????? ?????? ??? ???. ?????? 4(8). ?????????, 2011
2. q -????????????? [a, b]q = 0 ??? ab = qba,
[t?2? ?1 , t?1? ?2 ] =
?
t ? t ? ,
(?1 , ?2 )2 ?1 ?1 ?2 ?2
?
t ? t? ? ,
(?1 , ?2 )2 ?1 ?1 2 2
[t?1 ?1 , t?1 ?2 ]q = 0,
[t?1 ?1 , t?1 ?2? ]q = 0,
[t?1 ?2? , t?1 ?1? ]q = 0,
[t?2 ?2? , t?2 ?1? ]q = 0,
[t?2 ?1 , t?1? ?2 ] =
[t?1 ?1? , t?2 ?1? ]q = 0,
[t?1 ?1? , t?2? ?1? ]q = 0,
[t?2 ?2 , t?1? ?1? ] = (?1 , ?2 )2 ?t?1? ?2 t?2 ?1? ,
[t?1 ?2? , t?2 ?2? ]q = 0,
[t?1? ?2? , t?1? ?1? ]q = 0,
[t?1 ?1 , t?2 ?1 ]q = 0,
[t?1 ?1 , t?2? ?1 ]q = 0,
[t?2 ?1 , t?1? ?1 ]q = 0,
[t?2 ?2 , t?1? ?2 ]q = 0,
[t?1 ?2 , t?2 ?2 ]q = 0,
[t?2 ?1 , t?2 ?2 ]q = 0,
[t?2? ?1 , t?2? ?2 ]q = 0,
[t?1? ?1 , t?1? ?2? ]q = 0,
[t?2? ?1 , t?1? ?1 ]q = 0,
[t?2? ?2 , t?1? ?2 ]q = 0,
[t?1? ?1 , t?1? ?2 ]q = 0,
[t?2? ?1 , t?2? ?2? ]q = 0,
[t?2? ?2? , t?1? ?2? ]q = 0,
[t?2? ?1? , t?1? ?1? ]q = 0,
[t?2? ?2? , t?2? ?1? ]q = 0,
[t?2 ?1 , t?2 ?2? ]q = 0,
[t?1? ?2 , t?2? ?2? ]q = ?t?1? ?1 t?2? ?1? ,
[t?1? ?1 , t?2? ?1? ]q = 0,
[t?1? ?1 , t?2 ?1? ]q = 0,
[t?2 ?1? , t?2? ?2? ]q = ?t?1 ?1? t?1? ?2? ,
[t?1 ?2 , t?1? ?1 ]q = 0,
[t?1 ?2? , t?1? ?1 ]q = 0,
[t?1? ?2 , t?2 ?2? ]q = ?t?1? ?1 t?2 ?1? ,
[t?2 ?1 , t?1 ?1? ]q = 0,
[t?2? ?1 , t?1 ?1? ]q = 0,
[t?2? ?2 , t?1 ?2? ]q = ?t?2? ?1 t?1 ?1? ,
[t?1 ?1? , t?1? ?2 ]q = 0,
[t?1 ?1? , t?1? ?2? ]q = 0,
[t?2 ?2 , t?2 ?1? ]q = 0,
[t?2? ?2 , t?2? ?1? ]q = 0,
[t?1? ?2 , t?1? ?1? ]q = 0,
[t?2 ?2 , t?1? ?2 ]q = 0,
[t?2 ?2? , t?1? ?2? ]q = 0,
[t?2 ?1? , t?1? ?1? ]q = 0,
[t?1 ?2 , t?1 ?1? ]q = 0,
[t?2 ?2? , t?2 ?1? ]q = 0,
[t?2 ?1 , t?1? ?2? ] =
?
t ? t ?,
(?1 , ?2 )2 ?1 ?1 ?2 ?2
[t?1? ?2 , t?1 ?2? ] =
?
t ?t ? ,
(?1 , ?2 )2 ?1 ?1 ?1 ?1
?
t ?t ? ,
(?1 , ?2 )2 ?1 ?1 ?1 ?1
[t?2 ?2 , t?1 ?2? ]q = ?t?2 ?1 t?1 ?1? ,
[t?2 ?2 , t?2? ?1 ]q = ?t?1 ?2 t?1? ?1 ,
[t?1 ?2? , t?1? ?1? ]q = ?t?1 ?1? t?1? ?2? ,
[t?1 ?1 , t?1? ?2 ]q = ?t?1 ?2 t?1? ?1 ,
(25.b)
[t?1 ?2 , t?1? ?2 ]q2 = 0,
[t?1 ?2? , t?1? ?2? ]q2 = 0,
[t?1 ?1? , t?1? ?1? ]q2 = 0,
[t?1 ?1 , t?1 ?1? ]q2 = 0,
[t?2 ?1 , t?2 ?1? ]q2 = 0,
[t?1 ?1 , t?2 ?1? ]q = ?t?2 ?1 t?1 ?1? ,
[t?2 ?1? , t?2? ?2 ]q = ?t?1 ?1? t?1? ?2 ,
[t?2 ?2? , t?2? ?1 ]q = ?t?1 ?2? t?1? ?1 ,
[t?2 ?1 , t?1? ?1? ]q = ?t?1? ?1 t?2 ?1? ,
[t?1 ?1 , t?2? ?1? ]q = ?t?2? ?1 t?1 ?1? ,
(25.c)
[t?1 ?1 , t?1? ?2? ]q = ?t?1 ?2? t?1? ?1 ,
4. [a, b] = ?cd,
[t?2? ?1 , t?1? ?1? ]q = ?t?1? ?1 t?2? ?1? ,
[t?1 ?1 , t?2 ?2 ] = (?1 , ?2 )2 ?t?2 ?1 t?1 ?2 ,
[t?1 ?2 , t?1? ?1? ]q = ?t?1 ?1? t?1? ?2 ,
[t?1 ?1 , t?2? ?2? ] = (?1 , ?2 )2 ?t?2? ?1 t?1 ?2? ,
(25.e)
6. [a, b]q2 = ?cd,
[t?1 ?1 , t?2? ?2 ] = (?1 , ?2 )2 ?t?2? ?1 t?1 ?2 ,
[t?1 ?2 , t?2 ?1? ] =
?
t ? t? ? ,
(?1 , ?2 )2 ?1 ?1 2 2
[t?1 ?2 , t?1 ?2? ]q2 =
?
t? ? t ? ,
(?1 , ?2 )2 1 1 ?1 ?1
[t?1 ?2? , t?2? ?1? ] =
?
t ?t ? ?,
(?1 , ?2 )2 ?1 ?1 ?2 ?2
[t?2 ?1 , t?2? ?1 ]q2 =
?
t? ? t ? ,
(?1 , ?2 )2 1 1 ?1 ?1
[t?1 ?2? , t?2 ?1? ] =
?
t ?t ?,
(?1 , ?2 )2 ?1 ?1 ?2 ?2
[t?1? ?2 , t?1? ?2? ]q2 =
?
t ? t ? ?,
(?1 , ?2 )2 ?1 ?1 ?1 ?1
[t?1 ?1 , t?2 ?2? ] = (?1 , ?2 )2 ?t?2 ?1 t?1 ?2? ,
[t?2 ?2 , t?2? ?2 ]q2 = (?1 , ?2 )2 ?t?1 ?2 t?1? ?2 ,
?
t ?t ? ,
(?1 , ?2 )2 ?1 ?1 ?2 ?2
[t?2? ?2 , t?2? ?2? ]q2 = (?1 , ?2 )2 ?t?2? ?1 t?2? ?1? ,
[t?1 ?2 , t?2? ?1? ] =
(25.d)
5. [a, b]q = ?cd,
[t?1 ?1 , t?1? ?1 ]q2 = 0,
[t?1? ?1 , t?1? ?1? ]q2 = 0,
?
t ? t ? ?,
(?1 , ?2 )2 ?1 ?1 ?2 ?2
[t?2 ?1? , t?2? ?1 ] =
3. q 2 -????????????? [a, b]q2 = 0 ??? ab = q 2 ba,
[t?2? ?1 , t?2? ?1? ]q2 = 0,
[t?2? ?1 , t?1? ?2? ] =
[t?2 ?2 , t?2 ?2? ]q2 = (?1 , ?2 )2 ?t?2 ?1 t?2 ?1? ,
[t?2 ?2? , t?1? ?1? ] = (?1 , ?2 )2 ?t?1? ?2? t?2 ?1? ,
[t?2 ?2? , t?2? ?2? ]q2 = (?1 , ?2 )2 ?t?1 ?2? t?1? ?2? ,
[t?2? ?2 , t?1? ?1? ] = (?1 , ?2 )2 ?t?1? ?2 t?2? ?1? ,
[t?2 ?1? , t?2? ?1? ]q2 =
[t?2? ?2? , t?1? ?1? ] = (?1 , ?2 )2 ?t?1? ?2? t?2? ?1? ,
7
?
t ?t ? ?,
(?1 , ?2 )2 ?1 ?1 ?1 ?1
(25.f)
???????? ???? ???????? ?????? ??? ???. ?????? 4(8). ?????????, 2011
?qt?1 ?2? t?2 ?2 ? q 2 t?1 ?1? t?2 ?1 = 0,
7. ?????????
[t?1 ?1 , t?1? ?1? ]q2 = ?q(t?1 ?1? t?1? ?1 ? 1),
q ?2 t?1 ?1 t?2? ?1? + q ?1 t?1 ?2 t?2? ?2? ?
[t?2 ?2 , t?2? ?2? ] = ? (?1 , ?2 )2 t?1 ?2 t?1? ?2? +
?qt?1 ?2? t?2? ?2 ? q 2 t?1 ?1? t?2? ?1 = 0,
(
)
q ?2 t?2 ?1 t?1 ?1? + q ?1 t?2 ?2 t?1 ?2? ?
+(q + q ?1 )t?2? ?2 t?2 ?2? ,
(
)
[t?2 ?2? , t?2? ?2 ] = (?1 , ?2 )2 ? t?1 ?2? t?1? ?2 ? t?2? ?1 t?2 ?1? ,
[t?2 ?1 , t?2? ?1? ] =
(?1 , ?2 )2 q ?2 t?2 ?1 t?2 ?1? + q ?1 t?2 ?2 t?2 ?2? ?
?qt?2 ?2? t?2 ?2 ? (?1 , ?2 )2 q 2 t?2 ?1? t?2 ?1 = 0,
(
?
q ?2 t?1 ?1? t?1? ?1 +
(?1 , ?2 )2
?qt?2 ?2? t?1 ?2 ? q 2 t?2 ?1? t?1 ?1 = 0,
q ?2 t?2 ?1 t?1? ?1? + q ?1 t?2 ?2 t?1? ?2? ?
)
+(?1 , ?2 )2 (q + q ?1 )t?2 ?1? t?2? ?1 ,
?qt?2 ?2? t?1? ?2 ? q 2 t?2 ?1? t?1? ?1 = 0,
[t?1 ?2 , t?1? ?2? ] =
q ?2 t?2? ?1 t?1 ?1? + q ?1 t?2? ?2 t?1 ?2? ?
(
?
q ?2 t?1 ?1? t?1? ?1 +
(?1 , ?2 )2
?qt?2? ?2? t?1 ?2 ? q 2 t?2? ?1? t?1 ?1 = 0,
(?1 , ?2 )2 q ?2 t?2? ?1 t?2? ?1? + q ?1 t?2? ?2 t?2? ?2? ?
)
+(?1 , ?2 )2 (q + q ?1 )t?1? ?2 t?1 ?2? ,
(
?qt?2? ?2? t?2? ?2 ? (?1 , ?2 )2 q 2 t?2? ?1? t?2? ?1 = 0,
)
[t?2 ?1 , t?2? ?2 ]q = ? t?1 ?1 t?1? ?2 + qt?2? ?1 t?2 ?2 ,
q ?2 t?2? ?1 t?1? ?1? + q ?1 t?2? ?2 t?1? ?2? ?
[t?2? ?2 , t?1? ?2? ]q = ? t?2? ?2? t?1? ?2 + qt?2? ?1 t?1? ?1? ,
?qt?2? ?2? t?1? ?2 ? q 2 t?2? ?1? t?1? ?1 = 0,
(
)
(
)
q ?2 t?1? ?1 t?2 ?1? + q ?1 t?1? ?2 t?2 ?2? ?
[t?2 ?2? , t?2? ?1? ]q = ? t?1 ?2? t?1? ?1? + qt?2? ?2? t?2 ?1? ,
?qt?1? ?2? t?2 ?2 ? q 2 t?1? ?1? t?2 ?1 = 0,
)
(
[t?1 ?2 , t?2 ?2? ]q = ? qt?1 ?2? t?2 ?2 + t?1 ?1 t?2 ?1? ,
(
q ?2 t?1? ?1 t?1? ?1? + (?1 , ?2 )2 q ?1 t?1? ?2 t?1? ?2? ?
)
?(?1 , ?2 )2 qt?1? ?2? t?1? ?2 ? q 2 t?1? ?1? t?1? ?1 = 0,
[t?2 ?2 , t?2? ?1? ]q = ? t?1 ?2 t?1? ?1? + qt?2? ?2 t?2 ?1? ,
(
)
q ?2 t?1? ?1 t?2? ?1? + q ?1 t?1? ?2 t?2? ?2? ?
[t?2 ?2 , t?1? ?2? ]q = ? t?2 ?1 t?1? ?1? + qt?2 ?2? t?1? ?2 ,
)
(
?qt?1? ?2? t?2? ?2 ? q 2 t?1? ?1? t?2? ?1 = 0,
[t?1 ?2 , t?2? ?2? ]q = ? t?1 ?1 t?2? ?1? + qt?1 ?2? t?2? ?2 ,
(
? ????? ???????????? ????:
)
T t (j; ?)C ?1 (j)T (j; ?) = C ?1 (j),
[t?2 ?1 , t?2? ?2? ]q = ? t?1 ?1 t?1? ?2? + qt?2? ?1 t?2 ?2? .
?q ?2 t?1 ?1 t?1? ?1? ? (?1 , ?2 )2 q ?1 t?2 ?1 t?2? ?1? +
(25.g)
????? ????, ?????????? ????????????? ?????????????? ???????????? (22) ????:
+(?1 , ?2 )2 qt?2? ?1 t?2 ?1? + q 2 t?1? ?1 t?1 ?1? = ?q ?2 ,
t
?(?1 , ?2 )2 q ?2 t?1 ?2 t?1? ?2? ? q ?1 t?2 ?2 t?2? ?2? +
T (j; ?)C(j)T (j; ?) = C(j),
q ?2 t?1 ?1 t?1? ?1? + (?1 , ?2 )2 q ?1 t?1 ?2 t?1? ?2? ?
+qt?2? ?2 t?2 ?2? + (?1 , ?2 )2 q 2 t?1? ?2 t?1 ?2? = ?q ?1 ,
?(?1 , ?2 )2 qt?1 ?2? t?1? ?2 ? q 2 t?1 ?1? t?1? ?1 = q ?2 ,
?(?1 , ?2 )2 q ?2 t?1 ?2? t?1? ?2 ? q ?1 t?2 ?2? t?2? ?2 +
(?1 , ?2 )2 q ?2 t?2 ?1 t?2? ?1? + q ?1 t?2 ?2 t?2? ?2? ?
2 2
?qt?2 ?2? t?2? ?2 ? (?1 , ?2 ) q t?2 ?1? t?2? ?1 = q
2 ?2
(?1 , ?2 ) q
t?2? ?1 t?2 ?1? + q
?1
?1
+qt?2? ?2? t?2 ?2 + (?1 , ?2 )2 q 2 t?1? ?2? t?1 ?2 = q,
?q ?2 t?1 ?1? t?1? ?1 ? (?1 , ?2 )2 q ?1 t?2 ?1? t?2? ?1 +
,
t?2? ?2 t?2 ?2? ?
+(?1 , ?2 )2 qt?2? ?1? t?2 ?1 + q 2 t?1? ?1? t?1 ?1 = q 2 ,
?qt?2? ?2? t?2 ?2 ? (?1 , ?2 )2 q 2 t?2? ?1? t?2 ?1 = ?q,
?q ?2 t?1 ?1 t?1? ?1 ? (?1 , ?2 )2 q ?1 t?2 ?1 t?2? ?1 +
q ?2 t?1? ?1 t?1 ?1? + (?1 , ?2 )2 q ?1 t?1? ?2 t?1 ?2? ?
+(?1 , ?2 )2 qt?2? ?1 t?2 ?1 + q 2 t?1? ?1 t?1 ?1 = 0,
?(?1 , ?2 )2 qt?1? ?2? t?1 ?2 ? q 2 t?1? ?1? t?1 ?1 = ?q 2 ,
?q ?2 t?1 ?1 t?1? ?2 ? q ?1 t?2 ?1 t?2? ?2 +
q ?2 t?1 ?1 t?1 ?1? + (?1 , ?2 )2 q ?1 t?1 ?2 t?1 ?2? ?
+qt?2? ?1 t?2 ?2 + q 2 t?1? ?1 t?1 ?2 = 0,
?(?1 , ?2 )2 qt?1 ?2? t?1 ?2 ? q 2 t?1 ?1? t?1 ?1 = 0,
?q ?2 t?1 ?1 t?1? ?2? ? q ?1 t?2 ?1 t?2? ?2? +
q ?2 t?1 ?1 t?2 ?1? + q ?1 t?1 ?2 t?2 ?2? ?
+qt?2? ?1 t?2 ?2? + q 2 t?1? ?1 t?1 ?2? = 0,
8
(26)
???????? ???? ???????? ?????? ??? ???. ?????? 4(8). ?????????, 2011
?q ?2 t?1 ?2 t?1? ?1 ? q ?1 t?2 ?2 t?2? ?1 +
??? q = eJv , ? = q ? q ?1 = 2 sinh Jv . ????? ????????? ?????????? ???? ????? ??? j = ?, ??????????
??????? ????????? J = j 2 .
? ???????????? ? (20) ????? ?????????? ????????? ??????????????? ?????? ?????????? ?? ???????? j :
+qt?2? ?2 t?2 ?1 + q 2 t?1? ?2 t?1 ?1 = 0,
?(?1 , ?2 )2 q ?2 t?1 ?2 t?1? ?2 ? q ?1 t?2 ?2 t?2? ?2 +
+qt?2? ?2 t?2 ?2 + (?1 , ?2 )2 q 2 t?1? ?2 t?1 ?2 = 0,
t12 ? jt12 , t21 ? jt21 , t12? ? jt12? , t2? 1 ? jt2? 1 ,
?q ?2 t?1 ?2 t?1? ?1? ? q ?1 t?2 ?2 t?2? ?1? +
t21? ? jt21? , t1? 2 ? jt1? 2 , t2? 1? ? jt2? 1? , t1? 2? ? jt1? 2? ,
+qt?2? ?2 t?2 ?1? + q 2 t?1? ?2 t?1 ?1? = 0,
(29)
? ????????? ?? ??????????. ????? ???????? ?????????? ?????????? ??????, ??? ????????? ??????
Spv (2; j) ?? ??????? ?????? ?? ???????????, ?????????? ?? (25), ??????? ? ????? ???? ???????? ?????????????? ????????
?q ?2 t?1 ?2? t?1? ?1 ? q ?1 t?2 ?2? t?2? ?1 +
+qt?2? ?2? t?2 ?1 + q 2 t?1? ?2? t?1 ?1 = 0,
?(?1 , ?2 )2 q ?2 t?1 ?2? t?1? ?2? ? q ?1 t?2 ?2? t?2? ?2? +
[t11 , t22 ] = j 2 ?t21 t12 ,
[t11 , t33 ] = j 2 ?t31 t13 ,
[t11 , t32 ] = j 2 ?t31 t12 ,
[t12 , t24 ] =
?
t14 t22 ,
j2
?
t14 t33 ,
j2
[t13 , t24 ] =
?
t14 t23 ,
j2
[t11 , t23 ] = j 2 ?t21 t13 ,
[t12 , t34 ] =
?
t14 t32 ,
j2
?q ?2 t?1 ?1? t?1? ?1? ? (?1 , ?2 )2 q ?1 t?2 ?1? t?2? ?1? +
[t23 , t44 ] = j 2 ?t43 t24 ,
[t32 , t44 ] = j 2 ?t42 t34 ,
+(?1 , ?2 )2 qt?2? ?1? t?2 ?1? + q 2 t?1? ?1? t?1 ?1? = 0,
[t33 , t44 ] = j 2 ?t43 t34 ,
[t31 , t42 ] =
+qt?2? ?2? t?2 ?2? + (?1 , ?2 )2 q 2 t?1? ?2? t?1 ?2? = 0,
?q ?2 t?1 ?2? t?1? ?1? ? q ?1 t?2 ?2? t?2? ?1? +
+qt?2? ?2? t?2 ?1? + q 2 t?1? ?2? t?1 ?1? = 0,
?q
?2
t?1 ?1? t?1? ?2 ? q
?1
[t13 , t34 ] =
t?2 ?1? t?2? ?2 +
+qt?2? ?1? t?2 ?2 + q 2 t?1? ?1? t?1 ?2 = 0,
?q ?2 t?1 ?1? t?1? ?2? ? q ?1 t?2 ?1? t?2? ?2? +
+qt?2? ?1? t?2 ?2? + q 2 t?1? ?1? t?1 ?2? = 0.
(27)
?? ??? ??? ?????????????? ??????????? ??????????. ????? ?? ??? ???????? ? ?????????????? ???????????? ??? ??????????, ? ????????? ?????????
???????? ??????????? ?????????? ????????? ??????????????? ?????? Spv (2; j; ?).
??? n = 2 ?????? ???????????? S(2) ???????? ??? ????????: ????????????? ???????????? ?0 =
(1, 2) ? ???????????? ?? = (2, 1). ?????? ????????????? ???????? ?????? ?????????? ????????? ?????????? ? ???????????? ?????????? ?? ????????? j .
?????????? ??? ??????????? ????????.
3.1. ????????? ??????????????? ??????
Spv (2; j) ? ????????? ??????????????? ???????????? Sp4v (j)
? ?????? ????????????? ???????????? ?????
?????????? ????????? ???????????? ???????? Sp4v (j)
? ????????? ?????? ???????? Spv (2; j), ??????? ?0 ?
????????????. ???????, ??????????? ??? ???????,
?????????? ?? ?????? ??????????? ???????, ????
???????? ? ????????? ?k = k, k = 1, 2. ??? ???????? ?? Sp4q ? Sp4v (j) ?????????? x1 , x1? ?? ????????, ?
????????? ?????????? ?? j : x2 ? jx2 , x2? ? jx2? . ?
?????????? ?????????????? ??????????? (24) ????????? ? ????
x1 x2 = qx2 x1 ,
x1 x2? = qx2? x1 ,
x2? x1? = qx1? x2? ,
?
t41 t22 ,
j2
[t22 , t44 ] = j 2 ?t42 t24 ,
[t31 , t43 ] =
?
t41 t33 ,
j2
[t21 , t43 ] =
?
t41 t23 ,
j2
[t42 , t13 ] =
?
t14 t41 ,
j2
[t24 , t31 ] =
?
t14 t41 ,
j2
[t12 , t13 ]q2 =
?
t11 t14 ,
j2
[t21 , t31 ]q2 =
[t42 , t43 ]q2 =
?
t41 t44 ,
j2
[t22 , t32 ]q2 = j 2 ?t12 t42 ,
[t32 , t33 ]q2 = j 2 ?t31 t34 ,
[t22 , t23 ]q2 = j 2 ?t21 t24 ,
[t23 , t33 ]q2 = j 2 ?t13 t43 ,
[t24 , t34 ]q2 =
?
t11 t41 ,
j2
?
t14 t44 ,
j2
(
)
[t22 , t33 ] = ? j 2 t12 t43 + (q + q ?1 )t32 t23 ,
[t23 , t32 ] = j 2 ? (t13 t42 ? t31 t24 ) ,
(
[t21 , t34 ] =
[t12 , t43 ] =
)
?
q ?2 t14 t41 + j 2 (q + q ?1 )t24 t31 ,
j2
(
x2 x1? = qx1? x2 ,
?
x1 x1? ,
j2
[t21 , t42 ] =
t11 t44 ? q 2 t44 t11 = ?q(t14 t41 ? 1),
x1 x1? = q 2 x1? x1 ,
x2 x2? ? q 2 x2? x2 =
?
t41 t32 ,
j2
)
?
q ?2 t14 t41 + j 2 (q + q ?1 )t42 t13 .
j2
(30)
?????????????? ???????????, ??????? ?????????
?? (26),(27), ?? ?????????? ?? ?????.
(28)
9
???????? ???? ???????? ?????? ??? ???. ?????? 4(8). ?????????, 2011
3.3. ????????? ??????????????? ??????
3.2. ????????? ??????????????? ??????
Spv (2; j; ??) ? ????????? ???????????? Sp4v (j; ??)
Spv (2; ?) ? ????????? ??????????????? ???????????? Sp4v (?)
????? ?? ?????????? ????????? ????????????
? ????????? ??????, ?????????? ???????????? ?? =
(2, 1). ??? ????? ? ???????? ??????? 3 ??????? ?1 =
2, ?2 = 1. ????????? ??????????????? ????????????
Sp4v (j; ??) ??????? ?? ?????? (24). ??? ???????????
??????????? ? ??????????????? ?????????????
????????????????? ????????? ??????????????? ???????????? Sp4v (?) ? ????????? ??????????????? ?????? Spv (2; ?) ??????????, ???? ? ????????
??????????? ??????? ??????? ????????????? ???????? ????????? j = ?. ??? ???? ???????? ?????2
????? q = ej v |j=? = 1, ? j?2 = j22 sinh j 2 v|j=? = 2v .
????????? ??? ?????????? ??????????? ???????????? ?????????? ? ????, ?? ????? ?????????? ??????
???????? ?? ????.
????????????????? ????????? ??????????????? ???????????? ??????????????? ?????? ?????
????????? ????????????
Sp4v (?) = {[x2 , x2? ] = 2vx1 x1? } .
[x2 , x1 ]q = 0,
[x1? , x2? ]q = 0,
[t12? , t21? ] = 2vt11? t22? ,
[t2? 1 , t1? 2 ] = 2vt1? 1 t2? 2 ,
[t21 , t1? 2 ] = 2vt1? 1 t22 ,
[t1? 2 , t12? ] = 2vt11? t1? 1 ,
[t12 , t13 ] = 2vt11 t11? ,
[t1? 2 , t1? 2? ] = 2vt1? 1 t1? 1? ,
[t21 , t2? 1? ] = 2vt11? t1? 1 ,
(31)
[t22 , t11 ] = j 2 ?t12 t21 ,
[t12? , t2? 1? ] = 2vt11? t2? 2? ,
[t21? , t2? 1 ] = 2vt11? t1? 1 ,
[t21 , t12? ] =
?
t ?t ? ?,
j 2 22 1 1
[t21? , t1? 2? ] =
[t2? 1 , t1? 2? ] = 2vt1? 1 t2? 2? ,
[t22 , t11? ] = j 2 ?t12 t21? ,
[t21 , t1? 2? ] =
[t11? , t2? 2? ] = j 2 ?t2? 1? t12? ,
[t21 , t2? 1 ] = 2vt11 t1? 1 ,
[t1? 1? , t2? 2? ] = j 2 ?t2? 1? t1? 2? ,
[t21? , t2? 1? ] = 2vt11? t1? 1? ,
[t12 , t2? 1 ] =
(32)
t22 t2? 2? ? t22? t2? 2 = 1,
t11 t21? + t12 t22? ? t12? t22 ? t11? t21 = 0,
t11 t2? 1? + t12 t2? 2? ? t12? t2? 2 ? t11? t2? 1 = 0,
[t1? 2 , t2? 1? ] =
?
t ? t ? ?,
j2 2 2 1 1
[t2? 1 , t21? ] =
?
t ?t ? ,
j 2 22 2 2
[t21 , t21? ]q2 =
?
t22 t22? ,
j2
t2? 1 t1? 1? + t2? 2 t1? 2? ? t2? 2? t1? 2 ? t2? 1? t1? 1 = 0,
?
t ?t ? ,
j 2 22 1 1
[t1? 1 , t2? 2? ] = j 2 ?t2? 1 t1? 2? ,
[t1? 2 , t2? 1 ] =
?
t ? t ? ,
j2 2 2 1 1
[t11 , t2? 2? ] = j 2 ?t2? 1 t12? ,
?
t ? t ?,
j 2 2 2 11
[t12 , t2? 1? ] =
[t12? , t1? 2 ] =
?
t ?t ? ,
j 2 22 2 2
[t12 , t1? 2 ]q2 =
?
t22 t2? 2 ,
j2
[t11 , t1? 1 ]q2 = j 2 ?t21 t2? 1 ,
[t1? 1 , t1? 1? ]q2 = j 2 ?t1? 2 t1? 2? ,
[t11 , t11? ]q2 = j 2 ?t12 t12? ,
[t11? , t1? 1? ]q2 = j 2 ?t21? t2? 1? ,
?t11 t1? 2 ? t21 t2? 2 + t2? 1 t22 + t1? 1 t12 = 0,
?
t ?t ?,
j 2 22 11
?
t ? t ? ?,
j2 2 2 2 2
[t2? 1 , t2? 1? ]q2 =
t21 t1? 1? + t22 t1? 2? ? t22? t1? 2 ? t21? t1? 1 = 0,
?
t ? t11 ,
j2 2 2
?
t ? t11 ,
j 2 22
[t21? , t12? ] =
[t21 , t1? 2? ] = 2vt1? 1 t22? ,
(
[t12? , t1? 2? ]q2 =
?
t ?t ? ?,
j 2 22 2 2
)
[t11 , t1? 1? ] = ? j 2 t21 t2? 1? + (q + q ?1 )t1? 1 t11? ,
?t11 t1? 2? ? t21 t2? 2? + t2? 1 t22? + t1? 1 t12? = 0,
?t12 t1? 1? ? t22 t2? 1? + t2? 2 t21? + t1? 2 t11? = 0,
?t12? t1? 1? ? t22? t2? 1? + t2? 2? t21? + t1? 2? t11? = 0,
(34)
[t22 , t1? 1? ] = j 2 ?t1? 2 t21? ,
[t22 , t1? 1 ] = j 2 ?t1? 2 t21 ,
[t12 , t2? 1? ] = 2vt11? t2? 2 ,
[t12 , t1? 2? ] = 2vt11? t1? 1 .
q = exp(j 2 v),
?? ??????? ?????????????? ?????????????? ??????????? ??? ?????????? ????????? ??????????????? ?????? (??. ???? (35)) ????????, ??? J = j 2 .
????????? ????????, ??? ???????? ?????????? ????????????? ??? ??, ??? ? ? ?????? ????????????? ????????????. ? ?????????, ?? ?? ????? ??????????,
??? ???????? ????? ????? ??? ???????????? n.
????? ???????? ?????????? ?????????? ??????, ?? ??????? ?????? ?? ??????????? ?????????
??????????????? ?????? Spv (2; j; ??), ??????? ???????? ?????????????? ???????? ? ????? ????
????? ????, ?????????? ????????????? ?????????????? ????????????
t11 t1? 1? ? t11? t1? 1 = 1,
[x2 , x2? ]q2 = 0,
[x1 , x2? ]q = 0,
[x1 , x1? ]q2 = j 2 ?x2 x2? , ? = 2 sinh j 2 v.
??? ???????????? ????? ??????????????? ??????
???????????? ???????????????? ???????????? ? ????????????? ????? {x1 , x1? } ? ??????????????? ????? {x2 , x2? }.
????????? ??????????????? ?????? Spv (2; ?)
????? ????????? ????????? ?????????????? ??????????? ??????????
[t12 , t21? ] = 2vt11? t22 ,
[x2 , x1? ]q = 0,
(33)
? ??????? ??????? ?????????? ??????????? ?????????? ??????????? ?? ??????.
(
)
(
)
[t21 , t2? 1? ] =
?
q ?2 t22? t2? 2 + j 2 (q + q ?1 )t2? 1 t21? ,
j2
[t12 , t1? 2? ] =
?
q ?2 t22? t2? 2 + j 2 (q + q ?1 )t12? t1? 2 ,
j2
[t11? , t1? 1 ] = j 2 ? (t21? t2? 1 ? t1? 2 t12? ) .
10
(35)
???????? ???? ???????? ?????? ??? ???. ?????? 4(8). ?????????, 2011
Литература
????? ????, ?????????? ????????????? ?????????????? ???????????? (26),(27), ????? ?? ??????? ???????? ? ?????????????? ????????????
??? ??????????, ? ????????? ????????? ????????
??????????? ?????????? ????????? ??????????????? ?????? Spv (2; j; ??).
3.4. ????????? ??????????????? ??????
Spv (2; ?; ??) ? ????????? ???????????? Sp4v (?; ??)
??? ?????????? j = ? ???????? q = 1, ? = 2v ?,
??? ??? ??????? ?? (34), ???????????? Sp4v (?; ??) ?????????? ?????????????. ?????? ? ??? ????????? ?????? Spv (2; ?; ??) ????? ????????? ?????????????? ??????????? ??????????:
[t21 , t12? ] = 2vt22? t11 ,
[t21? , t1? 2? ] = 2vt22? t1? 1? ,
[t21? , t12? ] = 2vt22? t11? ,
[t21 , t1? 2? ] = 2vt22? t1? 1 ,
[t1? 2 , t2? 1 ] = 2vt2? 2 t1? 1 ,
[t1? 2 , t2? 1? ] = 2vt2? 2 t1? 1? ,
[t12 , t2? 1 ] = 2vt2? 2 t11 ,
[t12 , t2? 1? ] = 2vt2? 2 t11? ,
[t2? 1 , t21? ] = 2vt22? t2? 2 ,
[t12? , t1? 2 ] = 2vt22? t2? 2 ,
[t21 , t21? ]q2 = 2vt22 t22? ,
[t12 , t1? 2 ]q2 = 2vt22 t2? 2 ,
[t2? 1 , t2? 1? ]q2 = 2vt2? 2 t2? 2? ,
[t12 , t1? 2? ] = 2vt22? t2? 2 ,
1. Решетихин Н.Ю., Тахтаджян Л.А., Фаддеев
Л.Д. Квантование групп Ли и алгебр Ли // Алгебра и анализ, 1989. Т.1. Вып.1. С.178-206.
2. Gromov N.A., Kuratov V.V. Possible quantum
kinematics // J. Math. Phys., 2006. Vol. 47.
№ 1. P.013502-1-9.
3. Gromov N.A. Possible quantum kinematics. II.
Non-minimal case // J. Math. Phys., 2010. Vol.
51. № 8. P.083515-1-12; arXiv:1001.3978.
4. Inцnь E., Wigner E.P. On the contraction of
groups and their representations // Proc. Nat.
Acad. Sci. USA, 1953. Vol.39. P.510-524.
5. Громов Н.А. Контракции и аналитические продолжения классических групп. Единый подход. Сыктывкар: Коми НЦ УрО АН СССР,
1990. 220 с.
6. Gromov N.A. Contraction of algebraical structures and different couplings of Cayley-Klein
and Hopf structures // Turkish J. Phys., 1997.
Vol.3. P.377-383; q-alg/9602003.
7. Gromov N.A., Kostyakov I.V., Kuratov V.V. The
quantum symplectic Cayley-Klein groups // Int.
J. Mod. Phys. A, 1997. Vol. 12. № 1. P.177-182;
q-alg/9610010.
8. Gromov N.A., Kostyakov I.V., Kuratov V.V.
Quantum orthogonal Cayley-Klein groups in
Cartesian basis // Int. J. Mod. Phys. A, 1997.
Vol. 12. № 1. P.33-41; q-alg/9610011.
9. Громов Н.А., Костяков И.В., Куратов В.В.
Квантовые симплектические группы КэлиКлейна F un(Spq (n; j)) // Алгебра, дифференциальные уравнения и теория вероятностей.
Сыктывкар, 1997. С.30-43. (Труды Коми НЦ
УрО РАН, № 151).
10. Громов Н.А., Куратов В.В. Квантовые группы Кэли?Клейна SOv (N ; j; ?) в ортогональном
базисе // Алгебра, геометрия и дифференциальные уравнения. Сыктывкар, 2003. C.4-31.
(Труды Коми НЦ УрО РАН, № 174).
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groups // Ядерная физика, 2005. Т.68. № 10.
C.1752-1762; Phys. At. Nucl., 2005. Vol. 68. №
10. P.1689-1699.
[t12? , t1? 2? ]q2 = 2vt22? t2? 2? ,
[t21 , t2? 1? ] = 2vt22? t2? 2 .
(36)
????? ????, ?????????? ????????????? ?????????????? ???????????? ????:
t11 t21? ? t21 t11? + t12 t22? ? t12? t22 = 0,
t21 t1? 1? ? t21? t1? 1 + t22 t1? 2? ? t22? t1? 2 = 0,
t11 t2? 1? ? t11? t2? 1 + t12 t2? 2? ? t12? t2? 2 = 0,
t2? 1 t1? 1? ? t1? 1 t2? 1? + t2? 2 t1? 2? ? t2? 2? t1? 2 = 0,
t11 t1? 2 ? t12 t1? 1 + t21 t2? 2 ? t22 t2? 1 = 0,
t12 t1? 1? ? t11? t1? 2 + t22 t2? 1? ? t21? t2? 2 = 0,
t11 t1? 2? ? t12? t1? 1 + t21 t2? 2? ? t22? t2? 1 = 0,
t12? t1? 1? ? t11? t1? 2? + t22? t2? 1? ? t21? t2? 2? = 0,
t22 t2? 2? ? t22? t2? 2 = 1,
t11 t1? 1? ? qt11? t1? 1 = 1.
(37)
?????? ?????? ?????????, ??? ????????????????? ????????? ??????????????? ?????? ? ???????????????? ??????????? ????? ??????????? ??
????????????? ??????????????? ????????????, ? ??????? ?? ??????? 3.2, ??? ????????????????? ??????????????? ???????????? ??????????? ???????????????? ???????????.
?????? ????????? ? ???????? 22.03.2011.
11
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