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ADDENDUM TO ”ON SUPER-JORDANIAN sl N U ( ( |1)) h ALGEBRA” B. ABDESSELAMa, A. CHAKRABARTIb, R. CHAKRABARTIc, A. YANALLAHd and M.B ZAHAFe a,d,eLaboratoire de Physique Quantique de la Matière et Modélisations Mathématiques (LPQ3M), Centre Universitaire de Mascara, 29000-Mascara, Algérie aLaboratoire de Physique Théorique d’Oran, Université d’Oran Es-Sénia, 31100-Oran, Algérie bCentre de Physique Théorique, Ecole Polytechnique, 91128-Palaiseau cedex, France. 7 cDepartment of Theoretical Physics, University of Madras, Guindy Campus, Madras 600025, India 0 0 Abstract 2 We give a complete proof of the result (2.10) presented in our paper published in J. Phys. A: Math. n a Gen. 39 (2006) 8307–8319. J 1. Let U (sl(2|1)) (q is an arbitrary complex number)betheHopf superalgebragenerated by the elements 3 q 1 hˆ , eˆ and fˆ, i= 1,2, satisfy the relations i i i ] [hˆ , hˆ ]= 0, [hˆ , eˆ ] = a eˆ , [hˆ , fˆ]= −a fˆ, A i j i j ij j i j ij j Q qhî −q−hî [eˆ, fˆ] =δ , eˆ2 = fˆ2 = 0, . i j ij q−q−1 2 2 h t eˆ2eˆ − q+q−1 eˆ eˆ eˆ +eˆ eˆ2 = 0, fˆ2fˆ − q+q−1 fˆfˆfˆ +fˆfˆ2 = 0, (1) a 1 2 1 2 1 2 1 1 2 1 2 1 2 1 m (cid:16) (cid:17) (cid:16) (cid:17) [ where [ , ] is the supercommutator given by [a, b] = ab−(−)deg(a)deg(b)ba and a11 = 2, a12 = a21 = −1, a = 0. All generators are even except for eˆ and fˆ which are odd. The coproducts, counits and antipodes 1 22 2 2 v are given by 9 7 ∆(eˆ)= eˆ ⊗qhî/2+q−hî/2⊗eˆ, ǫ(eˆ) = 0, S(eˆ) = −qhî/2eˆq−hî/2, i i i i i i 3 1 ∆ fˆ = fˆ ⊗qhî/2+q−hî/2⊗fˆ, ǫ(fˆ)= 0, S(fˆ)= −qhî/2fˆq−hî/2, i i i i i i 0 (cid:16) (cid:17) 7 ∆ hˆ = hˆ ⊗1+1⊗hˆ , ǫ(hˆ )= 0, S(hˆ )= −hˆ . (2) i i i i i i 0 (cid:16) (cid:17) / h Let eˆ = eˆ eˆ −q−1eˆ eˆ and fˆ = fˆfˆ −qfˆfˆ. Khroshkin, Tolstoy [1] and Yamane [2] showed that 3 1 2 2 1 3 2 1 1 2 t a R = Rˆ K , (3) m q q q : where v i X K = q−h1⊗h2−h2⊗h1−2h2⊗h2, q ar Rˆ =exp q−q−1 q−h1/2eˆ ⊗fˆqh1/2 exp − q−q−1 q−h1/2−h2/2e ⊗fˆqh1/2+h2/2 × q q 1 1 q 3 3 h(cid:16) (cid:17) i h (cid:16) (cid:17) i exp − q−q−1 q−h2/2eˆ ⊗fˆqh2/2 (4) q 2 2 h (cid:16) (cid:17) i and exp (x) = xn/(n) !, (n) ! = (1) (2) ···(n) , (n) = 1−qk. In the (fund.)⊗(arbitrary) representa- q n≥0 q q q q q q 1−q tion, the R-matPrix (3) takes the following form: q−h2 A B   R | = 0 q−h1−h2 C , (5) q (fund.⊗arb.)        0 0 q−h1−2h2   where A = q−q−1 q−1/2fˆq−h1/2−h2, 1 (cid:16) (cid:17) 2 B = − q−q−1 q−1fˆqh1/2fˆq−h1−3h2/2− q−q−1 q−1/2fˆq−h1/2−3h2/2, 1 2 3 (cid:16) (cid:17) (cid:16) (cid:17) C = q−q−1 q−1/2fˆq−h1−3h2/2, (6) 2 (cid:16) (cid:17) 2. The R -matrix in the (fund.)⊗(arbitrary) representation is obtained, from (5), as follows: h G−1 − h G−1 0 G h G 0 q−1 q−1     R | = lim 0 G−1 0 R | 0 G 0 h (fund.⊗arb.) q→1  q (fund.⊗arb.)       0 0 G−1 0 0 G     G−1q−h2G α β   = lim 0 G−1q−h1−h2G γ , (7)   q→1     0 0 G−1q−h1−2h2   where h α = G−1q−h2G−G−1q−h1−h2G + q−q−1 q−1/2G−1fˆq−h1/2−h2G, 1 q−1(cid:16) (cid:17) (cid:16) (cid:17) 2 β = − q−q−1 q−1G−1fˆqh1/2fˆq−h1−3h2/2G− q−q−1 q−1/2G−1fˆq−h1/2−3h2/2G+ 1 2 3 (cid:16) (cid:17) (cid:16) (cid:17) h q−q−1 q−1/2G−1fˆq−h1−3h2/2G 2 q−1 (cid:16) (cid:17) γ = − q−q−1 q−1/2G−1fˆq−h1−3h2/2G (8) 2 (cid:16) (cid:17) and G= E h eˆ , E (x)= xn/[n]!, [n]!= [n]···[1], [n]= qn−q−n. q q−1 1 q n≥0 q−q−n (cid:16) (cid:17) 3. Defining P h h t(α) = E−1 eˆ E qα eˆ , t(0) = 1, (9) q (cid:18)q−1 1(cid:19) q(cid:18) q−1 1(cid:19) we obtain the following properties: heˆ heˆ E−1 1 qαh1/2E 1 = t(α)qαh1/2, (10) q (cid:18)q−1(cid:19) q(cid:18)q−1(cid:19) t(α+β)q(α+β)h1/2 = t(α)qαh1/2t(β)qβh1/2, (11) heˆ heˆ E−1 1 qβh2E 1 = t(−β)qβh2, (12) q (cid:18)q−1(cid:19) q(cid:18)q−1(cid:19) heˆ heˆ h E−1 1 fÊ 1 = fˆ − t(1)qh1 −t(−1)q−h1 , (13) q (cid:18)q−1(cid:19) 1 q(cid:18)q−1(cid:19) 1 (q−1)(q−q−1)(cid:16) (cid:17) heˆ heˆ E−1 1 fÊ 1 = fˆ, (14) q (cid:18)q−1(cid:19) 2 q(cid:18)q−1(cid:19) 2 heˆ heˆ hq E−1 1 fÊ 1 = fˆ + t(1)fˆqh1. (15) q (cid:18)q−1(cid:19) 3 q(cid:18)q−1(cid:19) 3 q−1 2 4. Let us introduce the following generator: T = limt(1). (16) q→1 From (11) it is evident that limt(α) = Tα. (17) q→1 To obtain a closed form of T, we proceed as follows: we left and right multiply the commutation relation qh1 −q−h1 = q−q−1 eˆ fˆ −fêˆ by G−1 and G respectively. After simple calculations, we reach to 1 1 1 1 (cid:0) (cid:1)(cid:16) (cid:17) t(2)qh1 −t(−2)q−h1 = qh1 −q−h1 +h(q+1) t(1)eˆ qh1 +q−h1eˆ t(−1) , (18) 1 1 h i which yields, when q −→ 1, to T2−T−2 = 2h T +T−1 eˆ ⇒ T −T−1 = 2heˆ . (19) 1 1 (cid:16) (cid:17) Finally, we obtain T±1 = ±heˆ + 1+h2eˆ2. (20) 1 1 q 5. We turn now to (7). It is easy to verify that limG−1q−h2G = limt(1)q−h2 = T, q→1 q→1 limG−1q−h1−h2G = limt(−1)q−h1−h2 = T−1, q→1 q→1 limG−1q−h1−2h2G = 1, q→1 limγ = −lim q−q−1 q−1/2fˆt(−1/2)q−h1−3h2/2 = 0, 2 q→1 q→1(cid:16) (cid:17) ht(1) ht(−1) limα = lim q−q−1 q−1/2fˆ + q−h2 −q−h1/2−h2−1/2 − q−h1−h2 −q−3h1/2−h2−1/2 1 q→1 q→1(cid:16) (cid:17) q−1(cid:16) (cid:17) q−1 (cid:16) (cid:17) h h = − T +T−1 h + T −T−1 1 2 2 (cid:16) (cid:17) (cid:16) (cid:17) h ≡ −hH + T −T−1 , 1 2 (cid:16) (cid:17) h limβ = −lim q−q−1 2q−1 fˆ − t(1)qh1 −t(−1)q−h1 t(1)qh/12fˆt(−1/2)q−h1−3h2/2 q→1 q→1(cid:16) (cid:17) (cid:20) 1 (q−1)(q−q−1)(cid:16) (cid:17)(cid:21) 2 hq − q−q−1 q−1/2 f + t(1)qh1 t(1/2)q−h1/2−3h2/2 3 (cid:16) (cid:17) (cid:20) q−1 (cid:21) h q−q−1 + q−1/2fˆt(−1/2)q−h1−3h2/2 (cid:0)q−1 (cid:1) 2 = 2h T −T−1 T1/2f −2hT3/2f +2hT−1/2f = 0. (21) 2 2 2 (cid:16) (cid:17) Finally, we obtain T −hH + h T −T−1 0 1 2  (cid:0) (cid:1)  R | = 0 T−1 0 . (22) h (fund.⊗fund.)          0 0 1   References [1] Khoroshkin S.M. and Tolstoy V.N., Comm. Math. Phys. 141 (1991), 599. [2] Yamane H., Publ. Res. Inst. Math. Sci. 30 (1994), 1587.