R-matrix Approach to Quantum Superalgebras suq(m | n)

arXiv:hep-th/9207075v1 22 Jul 1992NUHEP-TH-91-04R-matrix Approach to Quantum Superalgebras suq(m | n)D. Chang1, I. Phillips1, L. Rozansky1,21Department of Physics and Astronomy, Northwestern University,Evanston, Illinois 602082Department of Physics, Theory Group, RLM, University of Texas at Austin,Austin, Texas 78712AbstractQuantum superalgebras suq(m | n) are studied in the framework of R-matrixformalism. Explicit parametrization of L(+) and L(−) matrices in terms of suq(m |n) generators are presented.

We also show that quantum deformation of nonsim-ple superalgebra su(n | n) requires its extension to u(n | n).PACS: 02.40.+m, 05.50.+q

1IntroductionIn the course of studying quantum algebras, a lot of attention has been paid to the caseof quantum superalgebras (QSA) recently (see, for example, ref. [1], [2]).

These algebrasprovide solutions to the Yang-Baxter equation and therefore may serve as a source of newexactly solvable models in statistical mechanics [3]. It is also very interesting to study theirrelation to supergroups in the WZW models.

Such models were considered in ref. [4] andtheir connection with superconformal models was established.An intriguing relation between QSA and knot theory was discovered in ref.[5].

It wasshown there, that the QSA suq(n | n) is related to the Alexander-Conway polynomial inmuch the same way as the quantum algebra suq(n) is related to the Jones polynomial. Itstill remains to be seen how special properties of the Alexander-Conway polynomial arerelated to the nonsimplicity of the superalgebra su(n | n).QSA can also be shown to emerge from the “new” solutions to the Yang-Baxter equationdiscovered recently in a series of papers in ref.[6].

In those papers, these solutions were notrecognized as QSA because of the choice of parametrization of the matrices L(+) and L(−)in the framework of R-matrix formalism. With proper redefiniton of these parameters, QSAcan be easily demonstrated to be associated with these “new” solutions in very much thesame way the ”old” solutions are related to the usual quantum algebras.

Here we developa convenient parametrization of these matrices for the QSA suq(m | n).In contrast tothe previous papers on this subject [3, 2], we use ordinary- (instead of “super-”) R-matrices,which is in line with the approach of ref.[6]. Our treatment includes the case of nonsimpleQSA suq(n | n), the special features of which will be displayed.In following three sections we discuss the QSA suq(2 | 0), suq(0 | 2) and suq(1 | 1), whichare the building blocks of our general construction.

In section V we assemble these blocksin the matrices L(+) and L(−) for the QSA suq(m | n). In Appendix A we discuss possiblechoices of the R-matrix for that algebra, and in Appendix B we give a brief description ofthe QSA suq(2 | 1) as a specific example of our general result.1

2Quantum Superalgebra suq(2 | 0)Superalgebra suq(2 | 0) is, of course, the same as the algebra suq(2), which has been de-scribed, e.g. in ref.[7].

We repeat their analysis to establish notations that will make it easierto use as a building block for suq(m | n).According to the R-matrix method, one introduces the upper- and lower diagonal 2 × 2matrices L(+) and L(−). Their off-diagonal elements are raising and lowering generators ofsuq(2 | 0) while their diagonal elements are exponents of Cartan subalgebra generators.The basic commutation relations between the elements of L(+) and L(−) are expressedthrough the following relations[7]:L(±)2L(±)1= R21L(±)1 L(±)2R−121 , L(−)2 L(+)1= R21L(+)1 L(−)2 R−121(1)Here L(±)1= L(±) ⊗1, L(±)2= 1⊗L(±) and R21 = PRP, P is a permutation operator, so thatR21 = PRP =q1q −q−101q(2)Eq.

(1) implies the following commutation relations between the matrix elements of matricesL(±):L(±)11 L(+)12 = q∓1L(+)12 L(±)11 , L(±)22 L(+)12 = q±1L(+)12 L(±)22(3)L(±)11 L(−)21 = q±1L(−)21 L(±)11 , L(±)22 L(−)21 = q∓1L(−)21 L(±)22(4)hL(+)12 , L(−)21i= (q −q−1)L(+)11 L(−)22 −L(+)22 L(−)11(5)Eqs. (3) and (4) show that if L(+)12and L(−)21are proportional to the raising and loweringoperators X+ and X−, then both L(+)11and L(−)22 should be proportional to q−H, while L(+)22and L(−)11 should be proportional to qH, where H is a Cartan subalgebra element.

OperatorsX+, X−and H satisfy the standard commutation relations of suq(2):[H, X+] = X+,[H, X−] = −X−(6)2

[X+, X−] = q2H −q−2Hq −q−1(7)Our normalization for diagonal elements of matrices L(+) and L(−) differs from that of ref. [7]:L(±)11 = q∓12q∓H,L(±)22 = q∓12q±H(8)The advantage of such normalization is that these matrices have simple forms in the funda-mental representation of suq(2 | 0),L(±)11 = q∓1000,L(±)22 = q∓0001,(9)and can be easily generalized to the case of suq(m | n).To reconcile eqs.

(5) and (7), we have to introduce the factors (q −q−1) for L(+)12 and L(−)21as well as an extra negative sign which we ascribe to L(+)12 for reasons that we will explain inSection 5.Thus we arrive at the following parametrization of the matrices L(+) and L(−):L(+) =q−1000(q−1 −q)X+0q−0001(10)L(−) =q10000(q −q−1)X−q0001(11)where the matrices of the diagonal blocks should be interpreted in the sense of eq. (8) forarbitrary representations.3

3Quantum Algebra suq(0 | 2)The algebra suq(0 | 2) is, of course, isomorphic to suq(2 | 0). However the R-matrix that weshall use for this QSA is different from that of eq.

(2). Actually there are two choices forthis matrix in the literature:R21 =−q−1±1q −q−10±1−q−1(12)The upper signs are advocated in ref.

[6], while the lower ones - in ref.[2]. We will discussthe relation between these two possibilities in Appendix A.

Here we choose the lower signs,because, as it will be clear in Section 5, they simplify parametrization of matrices L(+) andL(−) in terms of QSA generators for the general case.With the choice of lower signs in eq. (12), commutation relations between matrix elementsof L(+) and L(−), which follow from eq.

(1), are:L(±)11 L(+)12 = q±1L(+)12 L(±)11 , L(±)22 L(+)12 = q∓1L(+)12 L(±)22(13)L(±)11 L(−)21 = q∓1L(−)21 L(±)11 , L(±)22 L(−)21 = q±1L(−)21 L(±)22(14)hL(+)12 , L(−)21i= (q−1 −q)L(+)11 L(−)22 −L(+)22 L(−)11(15)As expected this is nothing but eqs. (3, 4, 5) with q and q−1 interchanged.

This time eq. (13)and (14) show that if L(+)12and L(−)21are proportional to the raising and lowering operatorsY + and Y −, then both L(+)11and L(−)22should be proportional to qJ, while L(−)11and L(+)22should be proportional to q−J, where J is a Cartan subalgebra element.

As in the case ofsuq(2 | 0), operators Y +, Y −and J satisfy commutation relations of suq(2):[J, Y +] = Y +,[J, Y −] = −Y −(16)[Y +, Y −] = q2J −q−2Jq −q−1(17)4

Convenient normalization for diagonal elements of L+ and L−isL(±)11 = q± 12q±J = q±1000, L(±)22 = q± 12q∓J = q±0001,(18)where the 2 × 2 matrices are in the fundamental representation.We multiply all matrix elements of L(+) and L(−) by a factor (−1)F for future conve-nience. Here F is a fermionic number operator with eigenvalues 12 and −12 respectively in thefundamental representations of suq(2 | 0) and suq(0 | 2).

Obviously, an extra factor of (−1)Fdoes not affect the commutation relations (13-17), because it commutes with all operatorsinvolved. Thus we get the following parametrization of L(+) and L(−):L(+) =(−1)Fq1000(q −q−1)(−1)FY +0(−1)Fq0001(19)L(−) =(−1)Fq−10000(q−1 −q)(−1)FY −(−1)Fq−0001(20)where the diagonal blocks should be similarly interpreted as functions of the Cartan subal-gebra as in eqs.

(10, 11). For reasons that will become clear in Section 5, we give an extranegative sign to L(−)215

4Quantum Algebra suq(1 | 1)We begin, as usual, by presenting the matrix R21 for this algebra:R21 =q1q −q−101−q−1(21)The corresponding commutation relations for the matrix elements of L(+) and L(−) are(L(+)12 )2 = (L(−)21 )2 = 0(22)L(±)11 L(+)12 = q∓1L(+)12 L(±)11 , L(±)22 L(+)12 = −q∓1L(+)12 L(±)22(23)L(±)11 L(−)21 = q±1L(−)21 L(±)11 , L(±)22 L(−)21 = −q±1L(−)21 L(±)22(24)hL(+)12 , L(−)21i= (q −q−1)L(+)11 L(−)22 −L(+)22 L(−)11(25)Algebra suq(1 | 1) includes three generators Z+, Z−and E with (anti-)commutationrelations[E, Z+] = [E, Z−] = 0(26){Z+, Z−} = q2E −q−2Eq −q−1(27)In the fundamental representation E = I2, where I is identity operator.Since the operator E commutes with all generators of suq(1 | 1), it is clear that this setof operators is not enough to satisfy eqs. (23) and (24).

This deficit is a reflection of thedegeneracy of the Killing scalar product in the superalgebra su(1 | 1):StrEZ+ = StrEZ−= StrE2 = 0(28)In order to resolve these difficulties, we introduce another generator which we shall iden-tify with the fermion number operator F. It has eigenvalues 12 and −12 in the fundamentalrepresentation of suq(1 | 1). The operator F pairs with E in the Killing scalar product:Str EF = 12(29)6

and thus removes the degeneracy. It has the following commutation relations with othergenerators:[F, Z+] = Z+, [F, Z−] = −Z−, [F, E] = 0(30)If we now choose L(+)12 and L(−)21 to be proportional respectively to Z+ and Z−, then we shouldsetL(±)11 = q∓(F +E) = q∓1000, L(±)22 = (−1)Fq∓(F −E) = (−1)Fq±0001,(31)The factor (−1)F in the second expression is responsible for negative sign in the secondformulas of eqs.

(23) and (24). A factor of (−1)F added to the expression for L(+)12 will turnthe commutator (25) into the anticommutator (27).

Thus a complete parametrization ofL(+) and L(−) isL(+) =q−1000(q −q−1)(−1)FZ+0(−1)Fq0001(32)L(−) =q10000(q−1 −q)Z−(−1)Fq−0001(33)Inclusion of an additional generator F means that we actually produce the quantum defor-mation of nonspecial superalgebra uq(1 | 1) rather than suq(1 | 1). The same happens toother nonsimple superalgebras suq(n | n) which we discuss in the next section.In ref.

[8], anticommutation relations appear as a result of a graded tensor product,whereas in refs. [3, 2], the anticommutation relations appear through the use of “super-” Rmatrices.

Here, the ordinary-R matrices and the ordinary tensor product is used, but theparametrization of the L± matrices is supplemented by extra factors of (−1)F.7

5Quantum Superalgebra suq(m | n)The possible R-matrices for QSA suq(m | n) are discussed in Appendix A. Our choice ofmatrix R21 is a combination of matrices (2), (12) and (21).

This means that if we wantto permute two vectors in the fundamental representation, then we simply use one of thesematrices depending on whether both vectors are bosonic, fermionic, or one is bosonic andthe other is fermionic. The corresponding R-matrix in Eq.

(2) isR =XI(−1)pIq1−2pIeII ⊗eII +XI̸=J(−1)pIpJeII ⊗eJJ + (q −q−1)XI>JeIJ ⊗eJI(34)In our notations eIJ is an (m+n)×(m+n) matrix with only the (I, J) matrix element beingequal to 1, all other matrix elements are zero. We also set pI = 0 for bosons and pI = 1for fermions.

We will use indices I, J, . .

. for all (m + n) components of the fundamentalrepresentation, indices i, j, .

. .

only for the m bosonic variables and indices α, β, . .

. only forthe n fermionic ones.We compose matrices L(+) and L(−) out of the matrices (10), (11), (19), (20), (32) and(33) in the same way as we composed matrix R21:L(+)ii= q−eii,L(+)αα = (−1)Fqeαα,L(+)iα = (q −q−1)(−1)FZ+iα,L(+)ij= (q−1 −q)X+ij,i < j(35)L(+)αβ = (q −q−1)(−1)FY +αβα < βL(+)IJ = 0,I > JL(−)ii= qeii,L(−)αα = (−1)Fq−eαα,L(−)αi = (q−1 −q)Z−αi,L(−)ij= (q −q−1)X−ij,i > j(36)L(−)αβ = (q−1 −q)(−1)FY −αβα > βL(−)IJ = 0,I > JThe signs in front of the raising and the lowering operators X±, Y ± and Z± are chosenin such a way that when q →1, these operators tend to their classical counterparts.

This8

means that in the fundamental representation(X, Y, Z)±IJ →(Xcl, Ycl, Zcl)±IJ = eIJ(37)The generators of the classical superalgebra su(m | n) satisfy the (super-)commutationrelations[(Xcl, Ycl, Zcl)±IJ, (Xcl, Ycl, Zcl)±JK] = (Xcl, Ycl, Zcl)±IK(38)The corresponding relations for the matrix elements of L(+) and L(−) areL(+)IJ L(+)JK−(−1)PJ(PI+PK)L(+)JKL(+)IJ =(q−1 −q)(−1)PIPJL(+)JJ L(+)IK , (I < J < K)(39)L(−)IJ L(−)JK−(−1)PJ(PI+PK)L(−)JKL(−)IJ =(q −q−1)(−1)PKPJL(−)JJ L(−)IK , (I > J > K)(40)L(−)KI L(+)IJ−(−1)PI(PJ+PK)L(+)IJ L(−)KI =(q−1 −q)(−1)PIPKL(+)II L(−)KJ, (I < J < K)(41)L(−)KI L(+)IJ−(−1)PI(PJ+PK)L(+)IJ L(−)KI =(q −q−1)(−1)PIPJL(+)KJL(−)II , (I < K < J)(42)L(−)KJL(+)IK −(−1)PK(PI+PJ)L(+)IK L(−)KJ =(q−1 −q)(−1)PIPKL(+)IJ L(−)KK(I < J < K)(q −q−1)(−1)PJPKL(+)KKL(−)IJ(J < I < K)(43)L(−)JI L(+)JK = (−1)PKPI(−1)PJq1−2PJL(+)JKL(−)JI , (I < J < K)(44)L(−)KJL(+)IJ = (−1)PIPK(−1)PJq−1+2PJL(+)IJ L(−)KJ, (I < J < K)(45)Matching the classical limit of the eqs. (39-45) with the classical commutator (38) dictatesthe choice between factors (q −q−1) and (q−1 −q) for the operators X± and Y ±.

The signsof the operators Z± are not prescribed by these requirements and can be chosen arbitrarily.The exponents of q appearing in diagonal elements L(+)II and L(−)JJ are not supertraceless.Therefore, strictly speaking, they are not the elements of suq(m | n) Cartan subalgebra.To overcome this difficulty we multiply these matrix elements by factors of q1n−m and q1m−n9

respectively.Such factors will render the exponents supertraceless without affecting thecommutation relations (3-5), (13-15) and (23-25).This ends the process of parametrization of matrix elements of L(+) and L(−) in terms ofsuq(m | n) generators if m ̸= n. However supertracelesness can not be achieved if m = n. Inthis case the condition of supertracelesness of the original classical algebra su(n | n) should bedropped, so that we deal in fact with algebra u(n | n). Its Cartan subalgebra includes onegenerator with nonvanishing supertrace which can be identified with the fermion numberoperator F. If m ̸= n, F can be considered to be just an element of suq(m | n) Cartansubalgebra.6ConclusionWe considered the construction of quantum superalgebra suq(m | n) in the framework ofR-matrix formalism.

In contrast to the papers [2] and [3], we used ordinary (not super-)commutation relations between matrix elements of L(+) and L(−) while parametrizing themin terms of the generators of QSA suq(m | n). Thus it can easily be shown that the “special”solutions of Yang-Baxter equation, discussed in ref.

[6], are related to QSA in the same wayas ordinary solutions are related to quantum algebras through the Reshetikhin construction[9]. Therefore we conjecture that QSA can be used to generate all possible solutions to theYang-Baxter equation.Our study of nonsimple superalgebras su(n | n) also revealed that their quantum defor-mation requires extending them to superalgebras u(n | n), whose Cartan subalgebras includethe fermionic number operator.AcknowledgementIt is pleasure for us to acknowledge the stimulating conversations with Profs.

L. Kauffmanand H. Saleur on quantum superalgebras and Alexander polynomial. D.C. also wishes to10

thank the Institute of Physics at Academia Sinica for its hospitality while this manuscriptwas finalized. This work was supported in part by the U.S. Department of Energy.Appendix A – Choice of R-matrix for QSA suq(m | n)Here we discuss possible choices of R-matrix for QSA suq(m | n).

To simplify discussion,we will consider ˇR-matrix, which is the product of permutation operator P and originalR-matrix:ˇR = PR(A.1)The ˇR-matrix, presented in ref. [2], isˇR =XI̸=J(−1)pIpJeJI ⊗eIJ +XI(−1)pIq1−2pIeII ⊗eII + (q −q−1)XI

[6] for “nonstandard” solutions of the Yang-Baxter equation,has a slightly different form:ˇR′ =XI̸=JeJI ⊗eIJ +XI(−1)pIq1−2pIeII ⊗eII + (q −q−1)XI

(12) with lower signs, which stems from the ˇR-matrix in eq. (A.2), providessimple parametrization of L(+) and L(−) (the upper signs in eq.

(12) would correspond to theˇR-matrix (A.3)).To show the equivalence of the ˇR-matrices in eqs. (A.2) and (A.3), let us consider theaction of ˇR-matrix on the tensor productNNk=1 Vk of N fundamental representations Vk (Nis any integer).

Let us denote by eIk (1 ≤Ik ≤m + n) the basis vectors of Vk.Consider now operator D which calculates the parity of number of “fermionic disorders”in basis vectors of tensor product:DNOk=1eIk =NOk=1(−1)d{Ik}eIk(A.4)11

Here d{Ik} is the number of fermionic disorders, i.e. d{Ik} counts the number of pairs ofindices k, l, such thatk < l,m, Il, Ik(A.5)Obviously, D commutes with operators eII ⊗eJJ for all possible values of I and J, becausethese operators do not permute different vectors.

D commutes also with operators ejI ⊗eIj,because they permute pairs of vectors at least one of which is bosonic. HoweverDeαβ ⊗eβαD−1 = −eαβ ⊗eβα,α ̸= β(A.6)because operator eαβ ⊗eβα permutes two different fermionic vectors, thus changing the parityof fermionic disorder number.

Therefore we see thatDXI̸=J(−1)pIpJeJI ⊗eIJD−1 =XI̸=JeJI ⊗eIJ(A.7)andD ˇRD−1 = ˇR′(A.8)Eq. (A.8) shows equivalence of two ˇR-matrices (A.2) and (A.3).Appendix B – Quantum Superalgebra suq(2 | 1)As an example we give a brief description of the suq(2 | 1) quantum superalgebra.

Thematrices L± are parametrized as in eqs. (35),(36),L+ = q−1qh2(q−1 −q)α+(q −q−1)(−1)Fb+0qh1+h2(q −q−1)(−1)Fβ+00(−1)Fqh1+2h2(B.1)L−= qq−h200(q −q−1)α−q−h1−h20(q−1 −q)b−(q−1 −q)β−(−1)Fq−h1−2h2(B.2)12

Eqs. (1) and (34) implyb+α+ −qα+b+ = 0[β+, α+] = −b+qh1+h2[β−, α−] = b−q−(h1+h2)β+b+ + q−1b+β+ = 0β−b−+ q−1b−β−= 0α+b−−b−α+ = −β−qh2α+β−−qβ−α+ = 0α−b+ −b+α−= β+q−h2b+β−+ β−b+ = α+q−h1−2h2β+α−−q−1α−β+ = 0β+b−+ b−β+ = α−qh1+2h2.

(B.3)These relations reduce to the classical relations of su(2 | 1) in the classical limit. Adifferent parametrization can be found in Ref.

[2]AddendumAfter this paper was submitted, H. Saleur brought to our attention the paper ref. [8] whichaddresses the problem of suq(1 | 1).

The authors of ref. [8] use a graded tensor product inorder to produce anticommutators in some of the relations of (1).

We use an ordinary tensorproduct, which is in line with the approach taken in ref. [6].

However, we supplement theirparametrization of L± matrices with an extra factor of (−1)F to achieve the same effect.References[1] P. Kulish, N. Reshetikhin, Lett.Math.Phys. 18 (1989) 143.

[2] M. Chaichian, P. Kulish, Phys.Lett. 234B (1990) 72.

[3] S. Saleur, Nucl.Phys. B336 (1990) 363.

[4] M. Bershadsky, H. Ooguri, Phys.Lett. 229B (1989) 374.

[5] L. Kauffman, H. Saleur, Free Fermions and the Alexander-Conway Polynomial, preprintEFI 90-42.13

[6] N. Jing, M.-L. Ge, Y.-S. Wu, Lett.Math.Phys. 21 (1991) 193.Y.

Cheng, M.-L. Ge, K. Xue, New Solutions of Yang-Baxter Equation, preprint ITP-SB-90-38 (1990);[7] L. Faddeev, N. Reshetikhin, L. Takhtajan, Quantization of Lie Groups and Lie Algebras,preprint LOMI E-14-87 (1987)[8] Li-Liao, Xing-Chang Song, Mod.Phys.Lett. A6 (1991) 959.

[9] N. Reshetikhin, Quantized Universal Enveloping Algebras, the Yang-Baxter Equationand Invariants of Links, parts I and II, preprint LOMI E-4-87, E-17-87.14

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