2)

Chaos, Solitons and Fractals 11 (2000) 2123±2128

www.elsevier.nl/locate/chaos

Quantum group structure of Lie superalgebra osp…1=2† A. Hegazi *, M. Mansour Mathematics Department, Faculty of Science, Mansoura University, Mansoura 35516, Egypt Accepted 26 July 1999

Abstract In this paper we quantise the enveloping Lie superalgebra U…osp…1=2†† in the non-standard simple root system with two odd simple roots. The Quantum supergroup structure associated with the non-standard simple root system is also given. Ó 2000 Published by Elsevier Science Ltd. All rights reserved.

Quantum groups, quantum vector spaces and the underlying notion of deformations have substantially enriched the area of mathematics and mathematical physics. The algebraic structure of the Quantum group has been created by Jimbo [1] and Drinfeld [2]. The one-parameter Quantum groups are introduced as oneparameter deformation of the universal enveloping algebra U …L† of an algebra L leading to non-commutative and non-co-commutative Hopf algebra Uq …L† namely the Quantum group [7,8]. Quantum group would be also considered as a non-trivial generalization of the ordinary Lie groups. If L is a Lie algebra we deal with Quantum groups, and if L is a Lie superalgebra, we deal with a Quantum supergroups [3±6]. Let us consider the Lie superalgebra osp…1=2† whose even part is the Lie algebra sl…2† with the generators J3 , J and we call them isospin. The odd generators V are sl…2† spinors with the following commutation relations: ‰J3 ; J Š ˆ J ; ‰J ; V Š ˆ ÿV ; ‰J3 ; V Š ˆ 1=2V ; ‰V ; V Š ˆ J ; ‰J‡ ; Jÿ Š ˆ 2J3 ; ‰V‡ ; Vÿ Š ˆ J3 ; ‰J ; V Š ˆ 0: We will work over a ®eld K with char…K† ˆ 0; we assume that V‡2 ˆ f …q†J‡ ; Vÿ2 ˆ g…q†Jÿ ; ‰J3 ; V Š ˆ 1=2 V ; where f …q† and g…q† are functions of q and q 2 K  is generic. By direct way one can prove that our assumptions are consistent with the commutation relations of the Lie superalgebra osp…1=2†, that is ‰J3 ; J‡ Š ˆ J‡ ; ‰J3 ; Jÿ Š ˆ ÿJÿ ;

*

Corresponding author. E-mail address: [email protected] (A. Hegazi).

0960-0779/00/$ - see front matter Ó 2000 Published by Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 0 7 7 9 ( 9 9 ) 0 0 1 3 5 - 6

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consider also the following assumption: fV‡ ; Vÿ g ˆ F …J3 †; where F …J3 † is an arbitrary function of J3 . In this case we deduce that ‰J‡ ; Vÿ Š ˆ

1 fF …J3 ÿ 1=2† ÿ F …J3 †gV‡ ; f …q†

…1†

‰Jÿ ; V‡ Š ˆ

1 fF …J3 ‡ 1=2† ÿ F …J3 †gVÿ ; g…q†

…2†

‰J‡ ; Jÿ Š ˆ

1 ffF …J3 ÿ 1=2† ÿ F …J3 †gV‡ Vÿ ‡ fF …J3 † ÿ F …J3 ‡ 1=2†gVÿ V‡ g: f …q†g…q†

…3†

De®nition 1. We de®ne the following three types of q-number: ‰xŠ ˆ

qx=2 ÿ qÿx=2 ; q1=2 ÿ qÿ1=2

‰xŠq ˆ

qx ÿ qÿx ; q ÿ qÿ1

‰xŠ‡ ˆ

qx=2 ‡ qÿx=2 ; q1=2 ‡ qÿ1=2

with ‰xŠ‡ ‰xŠ ˆ ‰xŠq ; ‰xŠq ˆ ‰xŠqÿ1 and ‰xŠq ! x as q ! 1. If we choose F …J3 † ˆ ‰J3 Šq , then we obtain ‰J‡ ; Vÿ Š ˆ

ÿ1 ‰1=2Š‰2J3 ÿ 1=2Š‡ V‡ ; f …q†

…4†

‰Jÿ ; V‡ Š ˆ

1 ‰1=2Š‰2J3 ‡ 1=2Š‡ Vÿ ; g…q†

…5†

‰J‡ ; Jÿ Š ˆ

n o 1 2 ÿ ‰1=2Š‰2J3 ‡ 1=2Š‡ ‰J3 Šq ‡ …q1=4 ÿ qÿ1=4 † ‰J3 Šq V‡ Vÿ : f …q†g…q†

…6†

As q tends to 1 Eq. (4) becomes ‰J‡ ; Vÿ Š ˆ

ÿ1 V‡ ; 2f …1†

but

‰J‡ ; Vÿ Š ˆ ÿV‡ ;

this implies that f …1† ˆ 1=2. Also Eq. (5) becomes ‰Jÿ ; V‡ Š ˆ

1 Vÿ ; 2g…1†

but ‰Jÿ ; V‡ Š ˆ ÿVÿ then g…1† ˆ ÿ1=2:

We can choose f …q† and g…q† as follows: f …q† ˆ

1 ; q1=2 ‡ qÿ1=2

g…q† ˆ

ÿ1 q1=2 ‡ qÿ1=2

or any other form such that f …q† ! 1=2 as q ! 1 and g…q† ˆ ÿ1=2 as q ! 1.

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We get the following q-deformed osp…1=2† superalgebra: fV‡ ; Vÿ g ˆ ‰J3 Šq ;

…7†

‰J3 ; J Š ˆ J ;

…8†

‰J3 ; V Š ˆ 1=2V ;

…9†

‰J ; V Š ˆ 0;

…10†

‰V ; V Š ˆ J ;

…11†

‰J‡ ; Vÿ Š ˆ ÿ…q1=2 ‡ qÿ1=2 †‰1=2Š‰2J3 ÿ 1=2Š‡ V‡ ;

…12†

‰Jÿ ; V‡ Š ˆ ÿ…q1=2 ‡ qÿ1=2 †‰1=2Š‰2J3 ‡ 1=2Š‡ Vÿ ;

…13†

2

2

‰J‡ ; Jÿ Š ˆ ÿ…q1=2 ‡ qÿ1=2 † fÿ‰1=2Š‰2J3 ‡ 1=2Š‡ ‰J3 Šq ‡ …q1=4 ÿ qÿ1=4 † ‰J3 Šq V‡ Vÿ g:

…14†

De®ne the co-multiplication D, the co-unit e and the antipode s for Uq …osp…1=2†† as follows: D…qJ3 † ˆ qJ3 qJ3 ;

…15†

D…J3 † ˆ J3 1 ‡ 1 J3 ;

…16†

D…J † ˆ J qJ3 ‡ qÿJ3 J ;

…17†

D…V † ˆ V qJ3 ‡ qÿJ3 V ;

…18†

e…J † ˆ e…V † ˆ e…J3 † ˆ 0;

…19†

e…qJ3 † ˆ 1;

…20†

s…J † ˆ ÿq1 J ;

…21†

s…V † ˆ ÿq1 V ;

…22†

s…J3 † ˆ ÿJ3 ;

…23†

s…qJ3 † ˆ qJ3 :

…24†

Theorem 1. The q-deformed Lie superalgebra Uq …osp…1=2†† for jqj ˆ 1 under the co-multiplication D, the counit e and the antipode s defined by Eqs. (15)±(24) is a non-commutative and non-co-commutative Hopf superalgebra, subject to the following constraints: qJ3 qJ3 ˆ 1; qJ3 J qÿJ3 ˆ q1 J ; qJ3 V qÿJ3 ˆ q1 V :

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Proof. From the de®nition one can see that the Hopf superalgebra is associative, non-commutative and non-co-commutative. First, it is easy to see that the co-associativity of D (i.e., …I D†  D ˆ …D I†  D) and that the co-unit e is an algebraic homomorphism (i.e., e…ab† ˆ e…a†e…b†; a; b 2 Uq …osp…1=2††† with the co-unit property …I e†  D ˆ …e I†  D ˆ I. Now we will prove that D is an algebraic homomorphism i.e., D preserves relations (11)±(13). Thus,  qÿ1=2 ÿ 1 D qJ3 V‡ ‡ q1=2 qÿJ3 V‡ ÿ1 …q ÿ q †f …q† qÿ1=2 ÿ 1  J3 …q qJ3 †…V‡ qJ3 ‡ qÿJ3 V‡ † ˆ …q ÿ qÿ1 †f …q† ‡ q1=2 …qÿJ3 qÿJ3 †…V‡ qJ3 ‡ qÿJ3 V‡ † qÿ1=2 ÿ 1  J3 q V‡ 1 ‡ 1 qJ3 V‡ ˆ …q ÿ qÿ1 †f …q† ‡ q1=2 qÿJ3 V‡ 1 ‡ q1=2 1 qÿJ3 V‡ qÿ1=2 ÿ 1  J3 …Q V‡ ‡ q1=2 qÿJ3 V‡ † 1 ‡ 1 …qJ3 V‡ ‡ q1=2 qÿJ3 V‡ † ˆ ÿ1 …q ÿ q †f …q† ˆ ‰J‡ ; Vÿ Š 1 ‡ 1 ‰J‡ ; Vÿ Š:

D…‰J‡ ; Vÿ Š† ˆ

…i†

Also ‰DJ‡ ; DVÿ Š ˆ ‰J‡ qJ3 ‡ qÿJ3 J‡ ; Vÿ qJ3 ‡ qÿJ3 Vÿ Š ˆ ‰J‡ qJ3 ; Vÿ qJ3 Š ‡ ‰J‡ qJ3 ; qÿJ3 Vÿ Š ‡ ‰qÿJ3 J‡ ; Vÿ qJ3 Š ‡ ‰qÿJ3 J‡ ; qÿJ3 Vÿ Š but ‰J‡ qJ3 ; qÿJ3 Vÿ Š ˆ …J‡ qÿJ3 qJ3 Vÿ † ÿ …qÿJ3 J‡ Vÿ qJ3 † ˆ …qqÿJ3 J‡ qVÿ qJ3 † ÿ …qÿJ3 J‡ Vÿ qJ3 † ˆ 0: Similarly ‰qÿJ3 J‡ ; Vÿ qJ3 Š ˆ 0, ‰J‡ qJ3 ; Vÿ qJ3 Š ˆ J‡ Vÿ q2J3 ÿ Vÿ J‡ q2J3 ˆ …J‡ Vÿ ÿ Vÿ J‡ † 1 ˆ ‰J‡ ; Vÿ Š 1: Similarly ‰qÿJ3 J‡ ; qÿJ3 Vÿ Š ˆ 1 ‰J‡ ; Vÿ Š, then ‰DJ‡ ; DVÿ Š ˆ ‰J‡ ; Vÿ Š 1 ‡ 1 ‰J‡ ; Vÿ Š:

…ii†

From (i) and (ii) we get that D…‰J‡ ; Vÿ Š† ˆ ‰DJ‡ ; DVÿ Š. By the same way we can prove that D is an algebraic homomorphism for the other relations, and Uq …osp…1=2†† is bi-superalgebra. It remains to prove the equations of the antipode s. The antipode s is a z2 -graded vector space homomorphism s : Uq …osp…1=2†† ! Uq …osp…1=2†† satis®es M  …I s†  D ˆ M  …s I†  D ˆ u  e; M is the multiplication in Uq …osp…1=2†† and u is the unit u : K ! Uq …osp…1=2††. The antipode s can be extended to be an antialgebra, i.e., s…ab† ˆ …ÿ1†

jaj jbj

s…b†s…a†

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for any a; b 2 Uq …osp…1=2††, where jaj is the grade of the homogeneous element a, such that it satis®es (i) s  u ˆ u, (ii) e  s ˆ e, ‰aŠ‰bŠ (iii) T  …s s†  D ˆ D  s,where T is the twisting map de®ned by T …a b† ˆ …ÿ1† b a. (iv) s2 ˆ I: It is clear that s satis®es (i) and (ii). For (iii) we will check the relation for the elements qJ3 T  …s s†  D…qJ3 † ˆ T  …s s†…qJ3 qJ3 † ˆ T …qJ3 qJ3 † ˆ …qJ3 qJ3 † ˆ D…qJ3 † ˆ D…s…qJ3 †† ˆ D  s…qJ3 †: Similarly for the elements J and V . Let us now prove that s2 …‰J‡ ; Vÿ Š† ˆ ‰J‡ ; Vÿ Š: Thus,



qÿ1=2 ÿ 1  ÿ qÿ1 V‡ qÿJ3 ÿ qÿ1=2 qÿ1 V‡ qJ3 ÿ1 f …q†…q ÿ q †  qÿ1=2 ÿ 1 S qÿJ3 V‡ ‡ q1=2 qJ3 V‡ ˆ ÿ ÿ1 f …q†…q ÿ q † qÿ1=2 ÿ 1  ÿ1 J3 1=2 ÿ1 ÿJ3 V q ÿ q q V q ÿ q ˆ ÿ ‡ ‡ f …q†…q ÿ qÿ1 † qÿ1=2 ÿ 1  J3 q V‡ ‡ q1=2 qÿJ3 V‡ ˆ f …q†…q ÿ qÿ1 †



s2 …‰J‡ ; Vÿ Š† ˆ s

ˆ ‰J‡ ; Vÿ Š; we can of course prove this property for other elements of the superalgebra Uq …osp…1=2††. Now we will prove that the antipode s obeying M  …I s†  D ˆ M  …s I†  D ˆ u  e: Thus, M  …I s†  D…J † ˆ M  …I s†…J qJ3 ‡ qÿJ3 J † ˆ M…J qÿJ3 ÿ q1 qÿJ3 J † ˆ J qÿJ3 ÿ q1 qÿJ3 J ˆ q1 qÿJ3 J ÿ q1 qÿJ3 J ˆ 0; M  …s I†  D…J † ˆ M  …s I†…J qJ3 ‡ qÿJ3 J † ˆ M…ÿq J qJ3 ‡ qJ3 J † ˆ ÿ q1 J qJ3 ‡ qJ3 J ˆ ÿ q1 J qJ3 ‡ q1 J qJ3 ˆ 0; similarly for the other elements. Therefore Uq …osp…1=2†† is a non-commutative, non-co-commutative Hopf superalgebra, that is Uq …osp…1=2†† is Quantum supergroup and the proof is completed.

References [1] Jimbo M. Lett Math Phys 1986;11:247. [2] Drinfeld VG. In: Proceedings of International Congress of Mathematics, Berkeley, 1986. p. 798.

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