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ON A POSSIBLE ALGEBRA MORPHISM OF Uq[OSP(1/2N)] ONTO THE DEFORMED

arXiv:hep-th/9303142v1 25 Mar 1993ON A POSSIBLE ALGEBRA MORPHISM OF Uq[OSP(1/2N)] ONTO THE DEFORMEDOSCILLATOR ALGEBRA Wq(N)T. D. Palev* and N. I. Stoilova*International Centre for Theoretical Physics, 34100 Trieste, Italy*Abstract. We formulate a conjecture, stating that the algebra of n pairs of deformed Bose creationand annihilation operators is a factor-algebra of Uq[osp(1/2n)], considered as a Hopf algebra, andprove it for n = 2 case.

To this end we show that for any value of q Uq[osp(1/4)] can be viewed asa superalgebra, freely generated by two pairs B±1 , B±2 of deformed para-Bose operators. We writedown all Hopf algebra relations, an analogue of the Cartan-Weyl basis, the ”commutation” relationsbetween the generators and a basis in Uq[osp(1/2n)] entirely in terms of B±1 , B±2 .Mathematics Subject Classifications (1991).81R50, 16W30, 17B37.I.

IntroductionOne way to describe completely a given simple Lie (super)algebra A is in terms of its Chevalleygenerators. These generators are especially appropriate for a quantization of A, i.e., for a deformation ofthe universal enveloping algebra U[A] of A to a new associative algebra Uq[A] in such a way that Uq[A]remains a Hopf algebra.Another possible way to define A and U[A] was outlined in Ref.1.

It is based on the concept of creationand annihilation operators (CAO’s) of the simple Lie (super)algebra A under consideration. Contrary tothe Chevalley generators, the creation and annihilation operators of some algebras have direct physicalsignificance.

The CAO’s F ±1 , F ±2 , . .

. , F ±n of the orthogonal Lie algebra Bn ≡so(2n + 1) are known inquantum field theory as para-Fermi operators [2,3]; in a particular representation of Bn they become usualFermi operators.

Similarly, the CAO’s B±1 , B±2 , . .

. , B±n of the orthosymplectic Lie superalgebra osp(1/2n)are the para-Bose operators [5,6]; in the representation, corresponding to an order of the statistics p = 1they reduce to Bose creation and annihilation operators b±1 , b±2 , .

. .

, b±n .Clearly any deformation of U[osp(1/2n)] will lead to a deformation of B±1 , B±2 , . .

. , B±n and conse-quently to a deformation of the Bose operators b±1 , b±2 , .

. .

, b±n . In this relation we wish to rise and discusstwo questions.

* Permanent address: Institute for Nuclear Research and Nuclear Energy, 1784 Sofia, Bulgaria; E-mailaddress: palev@bgearn.bitnet1

1. Do the deformed CAO’s B±1 , B±2 , .

. .

, B±n of osp(1/2n) define entirely Uq[osp(1/2n)] (as this is thecase with the deformed Chevalley generators)?2. Is there any relation between the deformation of the Bose operators, obtained in this way, and theother known approaches to deform the Bose operators [7-11], which are unrelated to any Hopf algebrastructure?At present we do not know the answers neither to the first nor to the second question.

There are goodevidences, however, that the answer to both questions is positive and in particular that the Fock spacerepresentation of the q−deformed CAO’s B±1 , B±2 , . .

. , B±n coincides with the deformed Bose operators asdefined in Refs.8-10.In order to formulate our conjecture more precisely let Wq(n) be the deformed Weyl (or oscillator)algebra as defined by Hayashi [12].

The oscillator algebra Wq(n) is an associative algebra with unity, freegenerators b±i , k±i , i = 1, . .

. , n and the relations (i, j = 1, .

. .

, n)k−1iki = kik−1i= 1kib±i = q±1b±i kib−i b+i −q2b+i b−i = k−2i(1.2)b−i b+i −q−2b+i b−i = k2iaiaj = ajai,i ̸= j,where ai = b±i , k±1i.To turn Wq(n) into a superalgebra ( Z2-graded algebra) we setdeg(b±i ) = 1 ∈Z2,deg(k±1i) = 0 ∈Z2,i = 1, . .

. , n.(1.3)In the Fock representation of Wq(n) , namely when ki = qNi, b±i are the deformed q-bosons [8-10]and Ni is the ith boson number operator.

With respect to the grading following from (1.3) Wq(n) isan infinite-dimensional associative superalgebra. It is neither a Hopf algebra nor even a coalgebra.

Ourconjecture is the following one.CONJECTURE.The deformed Weyl superalgebra Wq(n) is a factor algebra of a deformed universalenveloping algebra Uq[osp(1/2n)].This conjecture holds in the nondeformed case [6]. In Sec.II we recall the idea of the proof.

At q ̸= 1the conjecture has been proved so far for n = 1 [13]. Here we prove it for n = 2.

To this end in Sec.III wedeform U[osp(1/4)] in terms of generators, which are in fact deformed para-Bose operators.The case n = 2 was considered in Ref.14 in relation to the ”supersingleton” Fock representation ofosp(1/4) and its ”singleton” [15] structure. The authors have studied in details the quantum deformationsof sp(4) at q being root of unity using deformed CAO’s.

We have been informed they have similar resultsto ours also for the deformed osp(1/4) [16].If the conjecture turns true, one can use the Hopf algebra structure in order to construct new repre-sentations of Uq[osp(1/2n)] or of any of its subalgebras beginning with some known representations of it2

and in particular with the representation ρF in the Fock space of the deformed Bose operators Fq(n) ≡Fockq(n). To this end one can use the comultiplication ∆.

For instance the maps∆(2) = (ρF ⊗ρF ) ◦∆: Uq[osp(1/2n)] →End(Fq(n) ⊗Fq(n))and∆(3) = (ρF ⊗ρF ⊗ρF ) ◦[(id ⊗∆) ◦∆] : Uq[osp(1/2n)] →End(Fq(n) ⊗Fq(n) ⊗Fq(n))(1.4)define representations of Uq[osp(1/2n)] in Fq(n) ⊗Fq(n) and Fq(n) ⊗Fq(n) ⊗Fq(n), respectively .Throughout we use the following abbreviations and notation:LS (LS’s)— Lie superalgebra (Lie superalgebras),lin.env. {X} — the linear envelope of X,Z — all integers,Z+ — all nonnegative integers,Z2 = (0, 1) — the ring of all integers modulo 2,[A, B] = AB −BA, {A, B} = AB + BA,< A, B >= AB −(−1)deg(A)deg(B)BA,[A, B]qn = AB −qnBA,{A, B}qn = AB + qnBA.24ptII.

The Nondeformed CaseLet Free(n) be the associative superalgebra with unity, free generatorsB±1 ,B±2 ,. .

. ,B±n ,deg(B±i ) = 1 and the relations (ξ, η, ǫ = ± or ±1, i, j, k = 1, 2, .

. .

, n)[{Bξi , Bηj }, Bǫk] = (ǫ −ξ)δikBηj + (ǫ −η)δjkBξi . (2.1)Consider the subspaceB(0/n) = lin.env.

{Bξi , Bηj }, Bǫk | ξ, η, ǫ = ±, i, j, k = 1, 2, . .

., n(2.2)and define a supercommutator on it< A, B >= AB −(−1)deg(A)deg(B)BA, A, B ∈B(0/n). (2.3)PROPOSITION 1 [6].

B(0/n) is a Lie superalgebra isomorphic to the orthosymplectic LS osp(1/2n)with an even subalgebrasp(2n) = lin.env. {Bξi , Bηj } | ξ, η, = ±, i, j = 1, 2, .

. ., n.(2.4)PROPOSITION 2.

The associative superalgebra Free(n) is (isomorphic to) the universal envelopingalgebra of osp(1/2n),Free(n) = U[osp(1/2n)]. (2.5)3

From these propositions one concludes that U[osp(1/2n)] is generated from B±1 , B±2 , . .

. , B±n∈osp(1/2n).

We point out that these 2n generators are very different from the Chevalley generators ofthe same algebra. The operators B±1 , B±2 , .

. .

, B±n were introduced by Green [2] as a possible general-ization of the Bose statistics and are called para-Bose operators. Propositions 1 and 2 indicate that therepresentation theory of the para-Bose statistics is simply another name for the representation theory ofthe orthosymplectic Lie superalgebra.Let b±1 , b±2 , .

. .

, b±n be Bose creation and annihilation operators and let W(n) be the correspondingWeyl superalgebra, i.e., the set of all polynomials of b±i , considered as odd variables. It is straightforwardto check that the Bose operators satisfy the para-Bose relations (2.1):[{bξi, bηj }, bǫk] = (ǫ −ξ)δikbηj + (ǫ −η)δjkbξi .

(2.6)This shows that the conjecture holds in the nondeformed case:PROPOSITION 3.The Weyl superalgebra W(n) is a factor-algebra of U[osp(1/2n)].Consequently any representation of W(n) and in particular its Fock representation is a representationof U[osp(1/2n)]. Hence one hasPROPOSITION 4.

The linear map ρ defined by the replacement B±i →b±i , i = 1, . .

. , n is a morphismof U[osp(1/2n)] onto W(n).The above proposition is in the origin of the so called ladder (or oscillator) representations .

From(2.4) and proposition 4 one obtainessp(2n) = lin.env. {bξi, bηj } | ξ, η, = ±, i, j = 1, 2, .

. .

, n.(2.7)Similarlygl(n) = lin.env. {b−i , b+j } | i, j = 1, 2, .

. .

, n.(2.8)III. The Superalgebra Uq[osp(1/4)]For a quantization of Uq[osp(1/4)] in terms of its Chevalley generators see Refs.17, 18.

Here weproceed in a different way, which will make it easier to prove the conjecture for n = 2 and is of independentinterest.Let Freeq(B±1 , B±2 , K±11 , K±12 ) be the associative algebra with unity, free generators B±1 , B±2 , K±11 ,K±12and the relations (ξ, η = ± or ±1)KiK−1i= K−1iKi = 1, K1K2 = K2K1, i = 1, 2,(3.1)Kξi Bηi = qξηBηi Kξi , Kξi Bηj = Bηj Kξi , i ̸= j = 1, 2,(3.2){B−i , B+i } = qK2i −q−1K−2iq −q−1, i = 1, 2,(3.3)4

[{Bξ1, Bη2}q−2ξη, yBǫ1]qη(ǫ−ξ) = 12(ǫ −ξ)q−η(ξ+1)(1 + q−2ξη)Bη2K−2η1,(3.4)[{Bξ1, Bη2}q−2ξη, Bǫ2]qξ(η−ǫ) = 12(ǫ −η)(1 + q2ξ)Bξ1K2ξ2 . (3.5)To turn Freeq(B±1 , B±2 , K±11 , K±12 ) into an associative superalgebra we setdeg(B±i ) = 1 ∈Z2, deg(K±1i) = 0 ∈Z2, i = 1, 2.

(3.6)PROPOSITION 5.Freeq(B±1 , B±2 , K±11 , K±12 ) is a Hopf superalgebra with a comultiplication ∆, acounit ǫ and an antipode S as follows:∆(B+1 ) = q−1/2B+1 ⊗K1K−22+ q−3/2K−11 K−22⊗B+1 + (q −q−1)q−1/2B+2 K−11 K−12⊗{B−2 , B+1 }q−2K−12 ,∆(B−1 ) = q3/2B−1 ⊗K1K22 + q1/2K−11 K22 ⊗B−1 + (q−1 −q)q1/2{B−1 , B+2 }q2K2 ⊗B−2 K1K2,∆(Bξ2) = q1/2Bξ2 ⊗K2 + q−1/2K−12⊗Bξ2, ξ = ±,∆(Kξi ) = qξ/2Kξi ⊗Kξi , i = 1, 2, ξ = ±1,(3.7)ǫ(Bξi ) = 0, ǫ(Kξi ) = q−ξ/2, i = 1, 2, ξ = ± or ± 1,S(Kξi ) = q−ξK−ξi, i = 1, 2, ξ = ±1,S(B+1 ) = (q7 −q5){B−2 , B+1 }q−2B+2 K22 −q7B+1 K42,S(B−1 ) = (q−7 −q−5)B−2 {B−1 , B+2 }q2K−22−q−7B−1 K−42 ,S(Bξ2) = −qξBξ2, ξ = ± or ± 1.PROPOSITION 6.The associative superalgebraFreeq(B±1 , B±2 , K±11 , K±12 )is a deformation ofU[osp(1/4)].For a proof set Ki = qHi−1/2 . Then as q →1 the relations (3.1)–(3.5) reduce to the para-Boserelations (2.1).

In particular Hi = 12{B−i , B+i }.In terms of the generators B±1 , B±2 , K±11 , K±12one can construct a q-analog of the Cartan-Weyl basis.For all values of q it is given with the following 14 generators:5

Ki, B±i , (B±i )2, {Bξ1, Bη2}q−2ξη, i = 1, 2, ξ, η = ± or ± 1. (3.8)PROPOSITION 7.

All ordered monomials (ni, mi ∈Z+, pi ∈Z)(B+2 )n1{B+1 , B+2 }n2q−2(B+1 )n3{B−2 , B+1 }n4q−2{B−1 , B+2 }m1q2 (B−1 )m2{B−1 , B−2 }m3q−2(B−2 )m4(K1)p1(K2)p2(3.9)constitute a basis in Uq[osp(1/4)].This is a q-deformed version of the Poincare-Birkhoff-Witt theorem. The proof follows from eqs.

(3.1)–(3.5) and the relations following from them, namely[{B−2 , B+1 }q−2, {B+1 , B+2 }q−2]q−2 = (1 + q−2)2(B+1 )2K22,[{B−2 , B+1 }q−2, {B−1 , B−2 }q−2]q2 = −(1 + q−2)(1 + q2)(B−2 )2K21,[{B−2 , B+1 }q−2, {B−1 , B+2 }q2] = q + q−1q −q−1 (K21K−22−K−21 K22),[{B−1 , B−2 }q−2, {B−1 , B+2 }q2]q2 = (1 + q−2)(1 + q2)(B−1 )2K−22 ,(3.10)[{B+1 , B+2 }q−2, {B−1 , B+2 }q2]q−2 = −(1 + q−2)2(B+2 )2K−21 ,[{B+1 , B+2 }q−2, {B−1 , B−2 }q−2] = 1 + q−21 −q2 (q2K21K22 −q−2K−21 K−22 ).Observe the very interesting situation that appears at q = ±i - a case which is not considered interms of the Chevalley basis [17, 18]. For these values of q the right hand sides of all equations (3.10)vanish.

This particular case deserves further investigations.IV. Proof of the Conjecture for n=2PROPOSITION 8.The Weyl superalgebra Wq(2) generated by the deformed Bose operators b±1 , k±11 ,b±2 , k±12is a factor-algebra of Uq[osp(1/4)].The proof is an immediate consequence of the observation that the deformed Bose operators b±1 , k±11 ,b±2 , k±12satisfy the defining relations for Uq[osp(1/4)].

More precisely, eqs. (3.1)–(3.5) remain valid afterthe replacement B±i →b±i , K±1i→k±1i.Consider the representation of Wq(2) in the Fock space Fockq(2) [8-10].

Then using proposition 8and eq. (1.4) we can write a representation of Uq[osp(1/4)] in Fockq(2) ⊗Fockq(2):∆(2)(B+1 ) = b+1 ⊗qN1−2N2−1/2 + q−N1−2N2−3/2 ⊗b+1 + (q1/2 −q−7/2)b+2 q−N1−N2 ⊗b+1 b−2 q−N2,6

∆(2)(B−1 ) = b−1 ⊗qN1+2N2+3/2 + q−N1+2N2+1/2 ⊗b−1 + (q−1/2 −q7/2)b−1 b+2 qN2 ⊗b−2 qN1+N2,∆(2)(Bξ2) = bξ2 ⊗qN2+1/2 + q−N2−1/2 ⊗bξ2, ξ = ±,(4.1)∆(2)(Kξi ) = qξ(Ni+1/2) ⊗qξNi, i = 1, 2, ξ = ±1.The operator ∆(2) defines a morphism of Uq[osp(1/4)] into Wq(2) ⊗Wq(2). In all essential points(as far as the representations of Uq[osp(1/4)] or of any of its subalgebras are concerned) ∆(2) is a goodsubstitute for a comultiplication in the Weyl algebra Wq(2).

The operator ∆(2) however does not satisfythe requirements for a comultiplication in Wq(2). In fact it is impossible to define a comultiplication inthe Weyl algebra even in the nondeformed case.Equations (4.1) indicate how to construct new representations of the superalgebra Uq[osp(1/4)] usingits oscillator representation, i.e., the Fock space representation of its factor-algebra Wq(2).

If the conjec-ture turns true, then the same approach can be applied for any Uq[osp(1/2n)]. Certainly, instead of theoscillator representation one can use any other representation.

The point is however that other represen-tations are at present unknown (contrary to the class of quantum superalgebras Uq[gl(n/1)] [19]). Thisstatement holds even for the ordinary, the nondeformed case and even for osp(1/4).

All finite-dimensionalrepresentations of the orthosymplectic LS’s osp(2m + 1/2n) are completely classified [4]. However (apartfrom osp(1/2) [20] and osp(3/2) [21]) explicit expressions for the transformations of the finite-dimensionalirreducible osp(2m + 1/2n) modules are not available.AckowledgmentsWe are thankful to Prof. Abdus Salam, the International Atomic Energy Agency and UNESCO forthe kind hospitality at the International Center for TheoreticalPhysics, where part of the present researchwas completed.

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