The q-commutators of braided groups
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1 206 ( ) Journal of East China Normal University (Natural Science) No. Jan. 206 : (206) q- (, 20024) : R-, [] ABCD U q(g).,, q-. : R- ; ; q- ; ; FRT- : O52.2 : A DOI: /j.issn The q-commutators of braided groups HU Hong-mei (Department of Mathematics, East China Normal University, Shanghai 20024, China) Abstract: With the standard R-matrices and suitably chosen a pair of dual braided groups, the authors gave the rank-inductive constructions of U q(g) for the ABCD series via the double-bosonization theory in []. This paper described explicitly the expressions for the generators of braided groups in the new higher rank-one quantum groups in these constructions, which are the q-commutators with the simple root vectors. These q-commutators are very important to the structure of new quantum groups. Key words: R-matrix; rank-inductive construction; q-commutator; braided group; FRT-generator 0 Majid Radford, [2-5]., Majid [6], FRT- [7], Drinfeld [8]. (H, R) M H ( H M)., C H M B M H, C H B U = U( C, H, B), H B C, H B C H : : (273) :,,,. hmhu024@26.com.
2 0 ( ) 206., g Q Q, Majid (H = kq = k[k ± i ], A = kq ), U q (n ± ) Cartan H, [9] U q (g)., H U q (g), (Uq ext (g), O q (G)), O q (G) ( ) V (R, R) V (R, R2 ) U q(g) M Uq(g) ( Uq(g)M), U(V (R, R2 ), U q(g), V (R, R)), []. V(R, R) V (R, R2 ), q-. ABCD [] ABCD. U ext q (g) [0], K m i, m Z,. O q (G), FRT- A(R) [7]. [6] Uq ext (g), O q (G), (m + ) i j, tk l = λrik jl, (m ) i j, tk l = (λr) ki lj, R R-,. λ, (m ± ) i j Uq ext (g) FRT-. Majid [] A(R) V(R, R) V (R, R. [] 2 ). R R-, R, : R 2 R 3 R 23 = R 23 R 3R 2, R 23 R 3 R 2 = R 2 R 3R 23 ; (PR+I)(PR I) = 0, R 2 R 2 = R 2 R 2. P. V (R, R) e i i =,, n}, e i e j = R ji ab ea e b. a,b e i t i a e a A(R) M,, (e i ) = e i + e i, ǫ(e i ) = 0, S(e i ) = e i, Ψ(e i e j ) = a,b R ji ab ea e b. f j, e i = δj i, V (R, R2 ) f j j =,, n}, f i f j = f b f a R ij ab. f i f a t a i MA(R), a,b, (f i ) = f i + f i, ǫ(f i ) = 0, S(f i ) = f i, Ψ(f i f j ) = a,b f b f a R ab ij. V (R, R) V (R, R2 ) Õq(G) = O q (G) k[g, g ], U ext (g) = U ext (g) k[c, c ]. q q, []..2 [] R ABCD R-. R A-, R = q 2 R; R BCD-, R = RPR (ǫq ǫ N+ + q 2 )R + (ǫq ǫ N+3 + )P (BD- ǫ =, C- ǫ = ). N, N = 2n; N, N = 2n +.
3 : q- U = (V (R, R2 ), Uext q (g), V (R, R)) e i (m + ) j k = λrji ab (m+ ) a ke b, (m ) i je k = λr ki abe a (m ) b j, (m + ) i jf k = λf b (m + ) i ar ab jk, () f i (m ) j k = λ(m ) j b f ar ab ik, [e i, f j ] = (m+ ) i j c c(m ) i j q q δ ij, (2) e i = e a (m + ) i a c + e i, f i = f i + c(m ) a i f a, (c) = c c. (3) () A- λ = q n+2, e, f, (m + ) c E, F K, U(V (R, R2 ), Uext q (sl ), V (R, R)) K ± i, i n, U q (sl n+2 ). B, C, D λ = q. (2) B- e 2, f 2, m + (m + ) 2 2 c E, F K, U(V (R, R2 ), Uext q (so 2 ), V (R, R)) U q (so 2n+3 ). (3) C- e 2n, f 2n, (m + ) 2n 2n c E, F K, U(V (R, R2 ), Uext q (sp 2n ), V (R, R)) K ± 2 U q (sp 2n+2 ). (4) D- e 2n, f 2n, (m + ) 2n 2n c E, F K, U(V (R, R2 ), Uext q (so 2n ), V (R, R)) K ± 2 K ± 2 2 K ± 2 K 2 2 U q (so 2n+2 ). V (R, R), V (R, R2 ).. V (R, R), V (R, R2 ) (R, R ). R, R. 2 q-.2 () (2). A. R- ( [9], [0]) R- P P. [x, y] q := xy qyx. 2. A- V (R, R 2 ) V (R, R) q- A- R- R ij kl = qqδij δ ik δ jl + (q 2 )δ il δ jk θ(j i), θ(k) =, k > 0, 0, k 0. (4) 2..2 () e, f E, F, V (R, R2 ) V (R, R)
4 2 ( ) 206 q-. e i = e i+ E i q E i e i+ = [e i+, E i ] q, (5) f i = qf i+ F i F i f i+ = q[f i+, F i ] q, i n. (6), U ext q (sl ) FRT- []. (m + ) i i+ = (q q )E i K (m ) i+ i = (q q )K (m + ) i i = K K 2 K 2 K 2 2 K i 2 K i 2 K i i K i i i K i i i K i i K (i+) (i+) K (n) i+ Kn, i n, i+ K (n) n F i, i n, (n) Kn, (m ) i i(m + ) i i =, i n +. (m + ) i i+ = (q q )E i (m + ) i+ i+, (m ) i+ i = (q q )(m ) i+ i+ F i. (7) V (R, R 2 ) V (R, R). e U q (sl n+2 ) E,.2 e i+ (m + ) i i+ = λr i a i+ b (m + ) a i+e b, f i+ (m ) i+ R- (4), i = λ(m ) i+ e i+ (m + ) i i+ = λq(m + ) i i+e i+ + λ(q 2 )(m + ) i+ i+ ei, b f a R ab i+ i, f i+ (m ) i+ i = (m ) i+ i+ f iλ(q 2 ) + (m ) i+ i f i+ λq, i n. (7), (q q )e i+ E i (m + ) i+ i+ = λq(q q )E i (m + ) i+ i+ ei+ + λ(q 2 )(m + ) i+ i+ ei, (8) (q q )f i+ (m ) i+ i+ F i = (m ) i+ i+ f iλ(q 2 ) + (q q )(m ) i+ i+ F if i+ λq. (9) e i+ (m + ) i+ i+ = λq2 (m + ) i+ i+ ei+, e i (m + ) i+ i+ = λq(m+ ) i+ i+ ei ; f i+ (m ) i+ i+ = (m ) i+ i+ f i+λq 2. (8), (9), (q q )e i+ E i (m + ) i+ i+ = λq(q q ) λq 2 E ie i+ (m + ) i+ i+ + λ(q2 ) λq ei (m + ) i+ i+, (0) (q q )λq 2 (m ) i+ i+ f i+f i = (m ) i+ i+ f iλ(q 2 ) + (q q )(m ) i+ i+ F if i+ λq. () (0), () q q (m ) i+ i+, λq(q q ) (m+ ) i+ i+, e i = e i+ E i q E i e i+ = [e i+, E i ] q, f i = qf i+ F i F i f i+ = q[f i+, F i ] q, i n.
5 : q- 3 (5) (6),. BCD. BCD R- kl = qqδji δ ji δ ik δ jl + (q 2 )θ(j l)(δ il δ jk K ij lk ). (2) R ij K ij lk = ǫci j Cl k, Cm t = ǫ m δ mt q ρm. g = so N, ǫ = ǫ = = ǫ N =, i < i, ρ i = N 2 i; i i, ρ i = ρ i. g = sp N, ǫ = = ǫ n =, ǫ = ǫ = = ǫ N =, i < i, ρ i = N 2 + i, ρ i = ρ i. i = N + i. B, C, D g Dynkin. < 2 > n n n n n n B n (n 2) C n (n 3) D n (n 4) 2.2 B- V (R, R 2 ) V (R, R) q- B-, ǫ = ǫ = = ǫ 2 =. i < n +, ρ i = 2 2 i; i = n +, ρ = 0; i > n +, ρ i = (i 2 2 ). (2) B- R-. K ij lk = Ci j Cl k = δ ij δ lk q ρi q ρ l, i + j 2n + 2, l + k 2n + 2, K ij lk = 0. i, j, R i,i+ i+,i = q2, R ii ii = qq δ ii = qq δi,2n+2 i = q 2, i n +, q, i = n +. R ij ij = qqδji δ ji = qq δj,2n+2 i = q, i + j 2n + 2, i j, i + j = 2n + 2, i j. q 2, i + j 2n + 2, i = j, 0, i + j 2n + 2, i j, R i,j i+,j = (q2 )(δ i,j δ j,i+ K i,j j,i+ ) = q 2 (q 2 ), i = n, j = n + 2, q 2 (q 2 ), i = j = n +, (q q ), i + j = 2n + 2, i j (2) e 2, f 2 U q (so 2n+3 ) E, F, V (R, R2 ) V (R, R)
6 4 ( ) 206 q-. e j = q E i e j e j E i = q [E i, e j ] q, i + j = 2n + 2, i n, (3) e = q 2 (q 2 + q 2 (q E e n+2 e n+2 E ) = q 2 (q 2 + q 2 [E, e n+2 ] q, (4) e n = (q 2 + q 2 (e E E e ), (5) e i = e i+ E i q E i e i+ = [e i+, E i ] q, i n ; (6) f j = F i f j qf j F i = [F i, f j ] q, i + j = 2n + 2, i n, (7) f = q 2 (q 2 + q 2 (F f n+2 qf n+2 F ) = q 2 (q 2 + q 2 [F, f n+2 ] q, (8) f n = (q 2 + q 2 (f F F f ), (9) f i = qf i+ F i F i f i+ = q[f i+, F i ] q, i n. (20) 2., V (R, R 2 ) V (R, R). (3) (6), (7) (20). U q (so 2 ), U ext q (so 2 ) FRT- []. (m + ) i i+ = (q q )E i K K 2 K n i, i n, (m + ) n = (q 2 + q 2 (q 2 q E, (m + ) i i = K K 2 K n i K i, (m + ) i i (m ) i i =, (m+ ) =, i n. (m + ) i i+ = (q q )E i (m + ) i+ i+, i n. i n, i + j = 2n + 2, n + 3 j 2n + n + 2 j 2n. R- e j (m + ) i i+ = λri,j a,b (m+ ) a i+ eb = λr i,j i,j (m+ ) i i+ ej + λr i,j i+,j (m+ ) i+ i+ ej = q (m + ) i i+ ej q (q q )(m + ) i+ i+ ej. (m + ) i i+ = (q q )E i (m + ) i+ i+ (q q )e j E i (m + ) i+ i+ = q (q q )E i (m + ) i+ i+ ej q (q q )(m + ) i+ i+ ej. (2) (2) e j (m + ) i+ i+ = λri+,j (m+ ) i+ i+ ej = (m + ) i+ i+ ej, e j (m + ) i+ i+ = λri+,j (m+ ) i+ i+ ej = q (m + ) i+ i+ ej, (q q )e j E i (m + ) i+ i+ = q (q q )E i e j (m + ) i+ i+ (q q )e j (m + ) i+ i+. (22) (22) q q (m ) i+ i+, ej = q E i e j e j E i, (3). e q-. e n+2 (m + ) n = λrn,n+2 (m+ ) n en+2 + λr n,n+2, (m+ ) e = q (m + ) n en+2 + q 2 (q q )e. (23)
7 : q- 5 (m + ) =, (m+ ) n = (q 2 + q 2 (q 2 q E (23), e = q 2 (q 2 + q 2 (q E e n+2 e n+2 E ), (4). e n, (24). e (m + ) n = λr n, n, (m+ ) n e + λr n,,n (m+ ) en = (m + ) n e + (q q )e. (24) (m + ) n = (q 2 + q 2) 2(q 2 q 2)E (24), e n = (q 2 + q 2) 2(e E E e ), (5)., i n, R-,. e i+ (m + ) i i+ = λri a i+ b (m + ) a i+ eb = λr i, i+ i, i+ (m+ ) i i+ ei+ + λr i,i+ i+,i (m+ ) i+ = (m + ) i i+ ei+ + (q q )(m + ) i+ i+ ei. (m + ) i i+ = (q q )E i (m + ) i+ i+, i+ ei (q q )e i+ E i (m + ) i+ i+ = (q q )E i (m + ) i+ i+ ei+ + (q q )(m + ) i+ i+ ei. (25) e i+ (m + ) i+ i+ = λri+,i+ i+,i+ (m+ ) i+ i+ ei+ = q(m + ) i+ i+ ej, (25) e i (m + ) i+ i+ = λri+,i (m+ ) i+ i+ ej = (m + ) i+ i+ ej, (q q )e i+ E i (m + ) i+ i+ = q (q q )E i e i+ (m + ) i+ i+ + (q q )e i (m + ) i+ i+. (26) (26) q q (m ) i+ i+, ei = e i+ E i q E i e i+. (6). V (R, R) U q (so 2n+3 ). 2.3 C- V (R, R 2 ) V (R, R) q- C- R-. g = sp 2n, ǫ = = ǫ n =, ǫ = ǫ = = ǫ 2n =. i n, ρ i = n + i; n + i 2n, ρ i = (i n). K ij lk. K ij lk = Cj icl k = ǫ iǫ l δ ij δ lk q ρi q ρ l = ǫ i ǫ l δ i,2 j δ l,2 k q ρi q ρ l ǫi ǫ l q ρi q ρ l, i + j = 2n +, l + k = 2n +, = 0,. i, Rii ii = qq δ ii = qq δi,2 i = q 2, R i,i+ i+,i = q 2 q 2, i = n, q 2, i n. q, i + j 2n +, i j, R ij ij = qqδji δ ji = qq δj,2 i =, i + j = 2n +. i + j = 2n +, j i +, R i, j i+, j : R i,j i+,j = (q2 )K i, j j, i+ = (q2 )ǫ i ǫ j q ρi q ρj = (q q ).
8 6 ( ) (3) e 2n, f 2n U q (sp 2n+2 ) E, F, V (R, R 2 ) V (R, R) q-. e j = q E i e j e j E i = q [E i, e j ] q, i + j = 2n +, i n, (27) e n = e E q 2 E e = [e, E ] q 2, (28) e i = e i+ E i q E i e i+ = [e i+, E i ] q, i n ; (29) f j = F i f j qf j F i = [F i, f j ] q, i + j = 2n +, i n, (30) f n = q 2 f F F f = q 2 [f, F ] q 2, (3) f i = qf i+ F i F i f i+ = q[f i+, F i ] q, i n. (32) U q (sp 2n ), Uq ext (sp 2n ) FRT- [],. (m + ) i i+ = (q q )E i (m + ) i+ i+, i n, (m + ) n = (q 2 q 2 )E K 2, (m + ) = K 2, (m + ) i i (m ) i i = (m ) i i (m+ ) i i =, (m+ ) i i (m+ ) i i =, (m ) i+ i = (q q )(m ) i+ i+ F i, i n, (m ) n = (q 2 q 2 )K 2 F. i + j = 2n +, i n, n + 2 j 2n n + j 2n. R-,.. R-, : e j (m + ) i i+ = q (m + ) i i+e j q (q q )(m + ) i+ i+ ej, e j (m + ) i+ i+ = (m+ ) i+ i+ ej, (33) (m + ) i i+ = (q q )E i (m + ) i+ i+, ej (m + ) i+ i+ = q (m + ) i+ i+ ej. (34), (27). (28), : e (m + ) n = q (m + ) n e + q (q 2 q 2 )(m + ) i+ i+ ej, (35) (m + ) n = (q 2 q 2 )E (m + ), e (m + ) = q(m+ ) e, (36) e n (m + ) = q (m + ) en. (37), R-,, : e i+ (m + ) i i+ = (m + ) i i+e i+ + (q q )(m + ) i+ i+ ei, e i (m + ) i+ i+ = (m+ ) i+ i+ ei, (38) (m + ) i i+ = (q q )E i (m + ) i+ i+, ei+ (m + ) i+ i+ = q(m+ ) i+ i+ ei+, (39) (29). (30) (32). 2.4 D- V (R, R 2 ) V (R, R) q-
9 : q- 7 g = so 2n, ǫ = ǫ = = ǫ 2n =. i n, ρ i = n i; ρ n+i = (i ). K ij lk = Ci j Cl k =δ i,2 jδ l,2 k q ρi q ρ l q ρi q ρ l, i + j=2n +, l + k=2n +, = 0,. 0, i = n, i, Rii ii = qq δ ii = qq δi,2 i = q 2. R i,i+ i+,i = q 2, i n. q, i + j 2n +, i j, R ij ij = qqδji δ ji = qq δj,2 i =, i + j = 2n +. i + j = 2n +, j i +, R i,j i+,j : R i,j i+,j = (q2 )K i,j j,i+ = (q2 )q ρi q ρj = (q q ) (4) e 2n, f 2n U q (so 2n+2 ) E, F, V (R, R 2 ) V (R, R) q-. e j = q E i e j e j E i = q [E i, e j ] q, i + j = 2n +, i n, (40) e n = q E e n+2 e n+2 E = q [E, e n+2 ] q, (4) e i = e i+ E i q E i e i+ = [e i+, E i ] q, i n ; (42) f j = F i f j qf j F i = [F i, f j ] q, i + j = 2n +, i n, (43) f n = F f n+2 qf n+2 F = [F, f n+2 ] q, (44) f i = qf i+ F i F i f i+ = q[f i+, F i ] q, i n. (45) U q (so 2n ), Uq ext (so 2n ) FRT- [], : (m + ) i i+ = (q q )E i (m + ) i+ i+, i n, (m + ) n n+2 = (q q )E (m + ) n+2 n+2, (m ) n+2 n = (q q )(m ) n+2 n+2 F, (m + ) i i(m ) i i = (m ) i i(m + ) i i =, (m + ) i i(m + ) i i =, (m ) i+ i = (q q )(m ) i+ i+ F i, i n. i + j = 2n +, i n, n + 2 j 2n n + j 2n. R-, : e j (m + ) i i+ = q (m + ) i i+e j q (q q )(m + ) i+ i+ ej, (46) (m + ) i i+ = (q q )E i (m + ) i+ i+, (47) e j (m + ) i+ i+ = (m+ ) i+ i+ ej, e j (m + ) i+ i+ = q (m + ) i+ i+ ej. (48), (40). e n,, R n,n+2, = 0. e n+2 (m + ) n n+2 = λr n,n+2 n,n+2 (m+ ) n n+2e n+2 + λr n,n+2, (m+ ) n+2 e λr n,n+2 n+2,n (m+ ) n+2 n+2 en = (m + ) n n+2e n+2 + (q q )(m + ) n+2 n+2 en.
10 8 ( ) 206 : (m + ) n = (q 2 q 2 )E (m + ), en+2 (m + ) n+2 n+2 = q(m+ ) n+2 n+2 en+2, (49) e n+2 (m + ) n n+2 = (m + ) n n+2e n+2 + (q q )(m + ) n+2 n+2 en, (50) e n (m + ) n+2 n+2 = (m+ ) n+2 n+2 en. (5), (4)., R-, : e i+ (m + ) i i+ = (m+ ) i i+ ei+ + (q q )(m + ) i+ i+ ei, (52) (m + ) i i+ = (q q )E i (m + ) i+ i+, (53) e i+ (m + ) i+ i+ = q(m+ ) i+ i+ ei+, e i (m + ) i+ i+ = (m+ ) i+ i+ ei. (54) (42). (43) (45). [ ] [ ] HU H M, HU N H. Double-bosonization and Majid s conjecture, (I): rank-induction of ABCD [J/OL]. Journal of Mathematics and Physics, 205, 56(702); [ 2 ] RADFORD D. Hopf algebras with projection[j]. Journal of Algebra, 985, 92(2): [ 3 ] RADFORD D, TOWBER J. Yetter-Drinfeld categories associated to an arbitrary bialgebra [J]. Journal of Pure and Appl Algebra, 993, 87(3): [ 4 ] MAJID S. Cross products by braided groups and bosonization [J]. Journal of Algebra, 994, 63(): [ 5 ] MAJID S. Some comments on bosonization and biproducts [J]. Czech Journal of Physics, 997, 47(2): 5-7. [ 6 ] MAJID S. Double-bosonization of braided groups and the construction of U q(g) [J]. Math Proceedings Cambridge Philos Society, 999, 25(): [ 7 ] FADDEEV L D, RESHETIKHIN N Y, TAKHTAJAN L A. Quantization of Lie groups and Lie algebras [J]. Leningrad Mathematics J, 990, (): [ 8 ] DRINFELD V G. Quantum groups [C]//Proceedings of the International Congress of Mathematicians. American Mathematics Society, 987, (2): [ 9 ] LUSZTIG G. Introduction to Quantum Groups [M]. Cambridge, MA: Birkhäuser, 993. [0] KLIMYK A, SCHMÜDGEN K. Quantum groups and their representations [M]. Berlin: Springer-Verlag, 997. [] MAJID S. Braided momentum in the q-poincare group. Journal of Mathematics Physics [J]. 993, 34(5): ( )
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