Cent. Eur. J. Phys. • 8(5) • 2010 • 819-824DOI: 10.2478/s11534-009-0168-8

Central European Journal of Physics

The multi-dimensional q-deformed bosonic Newtonoscillator and its inhomogeneous quantum invariancegroup

Research Article

Azmi A. Altıntaș1∗, Metin Arık2† , Ali S. Arıkan3‡

1 Okan University, Faculty of Engineering, Tuzla, Istanbul, Turkey

2 Bogaziçi University, Department of Physics, Bebek, Istanbul, Turkey

3 Sakarya University, Department of Physics, Esentepe, Sakarya, Turkey

Received 28 April 2009; accepted 17 November 2009

Abstract: We obtain the inhomogeneous invariance quantum group for the multi-dimensional q-deformed bosonicNewton oscillator algebra. The homogenous part of this quantum group is given by the multiparameterquantum group GLX ;qij (n) of Schirrmacher where qij ’s take some special values. We find the R-matrixwhich gives the non-commuting structure of the quantum group for the two dimensional case.

PACS (2008): 02.20.Uw

Keywords: bosonic Newton oscillator • quantum groups© Versita Sp. z o.o.

1. Introduction

Symmetry is an important concept in physics. Its impor-tance comes from not only its simplifying power but alsoits relation with conservation laws. Group theory, which isthe mathematical tool to study the symmetry of any phys-ical system, also has crucial importance in physics, espe-cially Lie groups and Lie algebras, which are very impor-tant in quantum theory. The deformation of Lie groups andLie algebras, which are necessary to quantize the classi-cal, completely integrable nonlinear systems, are known∗E-mail: ali.altintas@superonline.com†E-mail: metin.arik@boun.edu.tr‡E-mail: sarikan@sakarya.edu.tr (Corresponding author)

as quantum groups and quantum algebras [1–5]. Althoughthe most common way to define a quantum group is tomake a deformation of a Lie group, another way is bymaking transformations on an algebra which leaves thealgebra invariant and then check whether the transforma-tions satisfy the Hopf algebra axioms [6]. Since it has beenobserved that the quantum algebras can be constructed byusing deformed oscillator algebra, various studies [7, 8]were done related with the q-deformed oscillator algebra.One of them was defined by the following commutationrelations [9, 10]aia∗j − q2a∗j ai = q2Nδij , (1)aiN −Nai = ai, (2)

aiaj = ajai. (3)819

Author copy

The multi-dimensional q-deformed bosonic Newton oscillator and its inhomogeneous quantum invariance group

This system was called a multidimensional q-deformedbosonic Newton oscillator algebra, whose name comesfrom the fact that it was obtained from the standard quan-tum harmonic oscillator Newton equation. In these equa-tions, {ai} and {a∗i } are deformed annihilation and cre-ation operators respectively, where ∗ denotes hermitianconjugates. Here, q is the dimensionless quantity thatcan take only positive real values. N is the usual numberoperator and the subindexes, i and j , stand for 1 to d.One of the most important properties of the system de-fined by Eqs. (1) to (3) is related with its symmetry group.Although this algebra is a q-deformed one, its homoge-neous symmetry group, which represents rotations in theset of {ai} and the conjugate rotations in the set of {a∗i },is not a deformed Lie group. It is SU(d).In this study, we investigate whether finding an inhomo-geneous quantum symmetry group of the above system ispossible or not. It is obvious that such an investigation isquite difficult when the operator N is included. Therefore,we make a little change in the algebraic relations defin-ing a system, such that we write H instead of q2N andconsider the following systemaia∗j − q2a∗j ai = Hδij , (4)

aiH = q2Hai, (5)aiaj = ajai, (6)

and investigate the inhomogeneous quantum invariancegroup of the above algebraic structure. In the light ofEq. (2), one can write the following relationaif(N) = f(N + 1)ai (7)

for f(N) = ∑nk=1 fkNk . Thus, it is possible to realize thatEq. (5) comes from Eq. (2).In Sec. 2, we study the inhomogeneous invariance groupfor the two dimensional case and construct the R- ma-trix for this system. In Sec. 3, we generalize the resultsobtained in Sec. 2 for the d-dimensional case.

2. The inhomogeneous quantum in-variance group of the two dimen-sional bosonic Newton oscillatorFor the two dimensional consideration of the bosonicNewton oscillator system, the subindexes in Eqs. (4) to (6)count only 1 and 2. That is to say, one can write all rela-tions defining two dimensional bosonic Newton oscillator

as

a1a∗1 − q2a∗1a1 = H, (8)a2a∗2 − q2a∗2a2 = H, (9)

a1a∗2 = q2a∗2a1, (10)a1H = q2Ha1, (11)a2H = q2Ha2, (12)a1a2 = a2a1. (13)

In order to find its inhomogeneous invariance group, onecan write the following matrix M

M =

α11 α12 β11 β12 η1 γ1α21 α22 β21 β22 η2 γ2β∗11 β∗12 α∗11 α∗12 η∗1 γ∗1β∗21 β∗22 α∗21 α∗22 η∗2 γ∗20 0 0 0 χ3 χ40 0 0 0 0 1

, (14)

as the transformation matrix. The elements of transforma-tion matrix M are operators. χ3 and χ4 are hermitian oneswhereas αij , βij , γi and ηi are not hermitian for i, j = 1, 2.Their hermitian conjugates are denoted by using ∗ nota-tion. Construction of the above transformation matrix willbe meaningless unless the non zero elements of matrixM satisfy algebraic relations not only consistent amongthemselves but also belonging to a Hopf algebra [1–5, 11–16]. Before writing what these algebraic relations are, letus write the transformed form of the generators of the twodimensional Newton oscillator algebra

a′i = αik ⊗ ak + βik ⊗ a∗k + ηi ⊗H + γi ⊗ 1, (15)a∗′i = β∗ik ⊗ ak + α∗ik ⊗ a∗k + η∗i ⊗H + γ∗i ⊗ 1, (16)H ′ = χ3 ⊗H + χ4 ⊗ 1, (17)

by using Einstein summation convention. It is obvious thatsummation index k runs from 1 to 2 for the two dimensionalcase. Thus, one can find the following commutation rela-820

Author copy

Azmi A. Altıntaș, Metin Arık, Ali S. Arıkan

tions

αikαjl = αjlαik , (18)αikβjl = q−2βjlαik , (19)βikβjl = βjlβik , (20)αikβ∗jl = q2β∗jlαik , (21)αikα∗jl = α∗jlαik , (22)βikβ∗jl = q4β∗jlβik , (23)αikηj = q−2ηjαik , (24)αikγj = γjαik , (25)βikηj = q2ηjβik , (26)βikγj = γjβik , (27)ηiηj = ηjηi, (28)

ηiγj − γjηi = 12 (αjkβik − αikβjk ), (29)γiγj = γjγi, (30)αikχ3 = χ3αik , (31)αikχ4 = q2χ4αik , (32)βikχ3 = q4χ3βik , (33)βikχ4 = q2χ4βik , (34)ηiχ3 = q2χ3ηi, (35)ηiχ4 = q2χ4ηi, (36)γiχ3 = q2χ3γi, (37)γiχ4 = q2χ4γi, (38)αikη∗j = η∗j αik , (39)αikγ∗j = q2γ∗j αik , (40)βikη∗j = q4η∗j βik , (41)βikγ∗j = q2γ∗j βik , (42)ηiη∗j = q2η∗j ηi, (43)

ηiγ∗j − q2γ∗j ηi = 12 (χ3δij + q2β∗jkβik − αikα∗jk ), (44)γiγ∗j − q2γ∗j γi = χ4δij , (45)

and their hermitian conjugates. All of these relationsshould be satisfied in order to obtain the invariance ofthe Eqs. (8) to (13) according to the transformations (15)to (17).To say that successive application of transformations,identity transformation, and inverse transformation aremeaningful, the algebraic structure defined by the ele-ments of the transformation matrix M should be a Hopfalgebra [11]. The Hopf algebra structure of this systemcan be studied by defining coproduct, counit, and antipode

as∆(M) = M⊗M, (46)ε(M) = 1, (47)S(M) = M−1. (48)

Since the coproduct is defined by the matrix multiplication,one can write∆(αij ) = αik ⊗ αkj + βik ⊗ β∗kj , (49)∆(βij ) = αik ⊗ βkj + βik ⊗ α∗kj , (50)∆(ηi) = αik ⊗ ηk + βik ⊗ η∗k + ηi ⊗ χ3, (51)∆(γi) = αik ⊗ γk + βik ⊗ γ∗k + ηi ⊗ χ4 + γi ⊗ 1, (52)∆(χ3) = χ3 ⊗ χ3, (53)∆(χ4) = χ3 ⊗ χ4 + χ4 ⊗ 1. (54)

From algebra homomorphism, the above coproducts shouldalso preserve the relations (18) to (45). This requirementresults with the following additional commutation relationχ3χ4 = χ4χ3. (55)

Thus, the algebra satisfied by the elements of the trans-formation matrix M can be described by Eqs. (18) to (45)and (55), together with their hermitian conjugates. Al-though we have found all commutation relations satisfiedby the transformation matrix elements, we still have noidea about the inverse of matrix M. To find it, let usrewrite transformation matrix M in the following wayM = ( A Γ0 B

), (56)

where A is the 4× 4 square matrix, B is the 2× 2 squarematrix and Γ is the 4 × 2 matrix. With the help of theabove notation, it is not difficult to realize that the inverseof matrix M can be written asM−1 = ( A−1 −A−1ΓB−10 B−1

), (57)

since MM−1 = M−1M = 1 is satisfied. The elements ofmatrix B commute among themselves. Therefore, it is notdifficult to write the inverse of matrix B. But what aboutthe inverse of matrix A. Since the elements of matrix Asatisfy the Eqs. (18) to (23), it is a good idea to write thematrix A asA = ( α β

β∗ α∗

). (58)

821

Author copy

The multi-dimensional q-deformed bosonic Newton oscillator and its inhomogeneous quantum invariance group

Here α and β stand for representing the following 2 × 2square matricesα = ( α11 α12

α21 α22), (59)

β = ( β11 β12β21 β22

). (60)

α∗ and β∗ are also 2 × 2 square matrices. But their el-ements correspond to the hermitian conjugate of the ele-ments of α and β matrices respectively. The commutationrelations satisfied by α , β, α∗, and β∗ are the same asthe commutation relations written for GLq2,q−2 (2) [16, 17].Therefore, one can write the inverse of matrix A as [17]A−1 = ( α∗ −q2β

−q−2β∗ α

)D−1, (61)

where D is the following 2× 2 square matrixD = αα∗ − q−2ββ∗. (62)

Hence, it is possible to recognize that only one point re-mains to write the inverse of matrix A. This is finding D−1.In order to find D−1, let us rewrite matrix D asD = ( D11 D12

D21 D22), (63)

whereD11 = α11α∗11 + α12α∗21 − q−2β11β∗11 − q−2β12β∗21, (64)D12 = α11α∗12 + α12α∗22 − q−2β11β∗12 − q−2β12β∗22, (65)D21 = α21α∗11 + α22α∗21 − q−2β21β∗11 − q−2β22β∗21, (66)D22 = α21α∗12 + α22α∗22 − q−2β21β∗12 − q−2β22β∗22. (67)

By considering the commutation relations (18) to (23), onecan find that Dij ’s commute among themselves. Therefore,as in matrix B, it is not difficult to find the inverse of ma-trix D, which should be found in order to write the inverseof transformation matrix M.Up to now, we have discovered that the two dimensionalbosonic Newton oscillator algebra has a symmetry thatcan be described by a quantum group. This situationmakes it possible to write the R-matrix that stores up allinformation about the algebraic structure of the symmetry

group. To find the R-matrix, one can write the RMM-relation [17, 18]

RM1M2 = M2M1R, (68)

where M1 ≡ M⊗1 and M2 ≡ 1⊗M. Due to the definitionofM1, M2 and existence of the RMM-relation, it is obviousthat the R-matrix is a 36× 36 square matrix whose non-zero elements can be written as

R11 = R22 = R66 = R77 = R88 = R1212 = R1515= R1616 = q2, (69)R1818 = R2121 = R2222 = R2424 = R2929 = R3030 = R3131 = q2, (70)

R3232 = R3333 = R3434 = R3535 = R3636 = q2, (71)R33 = R44 = R55 = R99 = R1010 = R1111 = R2727= R2828 = 1, (72)

R1313 = R1414 = R1717 = R1919 = R2020 = R2323 = R2525= R2626 = q4, (73)R330 = R335 = R1030 = R1035 = −12 , (74)R1330 = R1335 = R2030 = R2035 = q22 . (75)

Here, the upper indices are used to represent the rownumber whereas the lower indices are used to representcolumn number of the matrix elements.3. The inhomogeneous quan-tum symmetry group for multi-dimensional bosonic Newton os-cillator algebra

Now, let us extend our study from two dimensional tothe d-dimensional one which is defined by the Eqs. (4)to (6). In these equations all indices can take the values 1to d. To find the invariance group of this system, one canwrite the transformation of the generators of this algebraicsystem with the help of (2d+ 2)× (2d+ 2) square matrix822

Author copy

Azmi A. Altıntaș, Metin Arık, Ali S. Arıkan

T , namely,

a′1a′2...a′da∗′1a∗′2...a∗′dH ′1′

= T

a1a2...ada∗1a∗2...a∗dH1

= ( A Υ0 B

)

a1a2...ada∗1a∗2...a∗dH1

. (76)

In the above equality, transformation matrix T was writtenin a shorthand notation. It includes 2d×2d square matrixA,

A =

α11 α12 · · · α1d β11 β12 · · · β1dα21 α22 · · · α2d β21 β22 · · · β2d... ... ... ... ... ... ... ...αd1 αd2 · · · αdd βd1 βd2 · · · βddβ∗11 β∗12 · · · β∗1d α∗11 α∗12 · · · α∗1dβ∗21 β∗22 · · · β∗2d α∗21 α∗22 · · · α∗2d... ... ... ... ... ... ... ...β∗d1 β∗d2 · · · β∗dd α∗d1 α∗d2 · · · α∗dd

, (77)

2× 2 square matrix BB = ( χ3 χ40 1

), (78)

and 2d× 2 matrix Υ

Υ =

η1 γ1η2 γ2... ...ηd γdη∗1 γ∗1η∗2 γ∗2... ...η∗d γ∗d

. (79)

By acting with this matrix T on the generators, one canfind the relations (18) to (45) and (55), which leave Eqs. (4)to (6) unchanged under this transformation. In this case,all indices in Eqs. (18) to (45) run 1 to d. Although, thisleads to having many more equations than in the two di-mensional case, the quantum group which represents the

symmetry of this system does not have a more complicatedstructure than the two dimensional case.In order to see the existence of inverse within the trans-formation matrix T , it is possible to follow similar steps asthose considered in the previous section. That is to say,one can writeT−1 = ( A−1 −A−1ΥB−10 B−1

), (80)

which concentrates to the matrix A whose elements donot commute among themselves. To verify the existence ofthe inverse of matrix A, it is also possible to consider themultiparametric deformation, GLpij ,qij (n), of GL(n) [19]. InSchirrmacher’s study [19], the commutation relations formultiparametric deformed quantum group GLpij ,qij (n) weregiven with the following equationsAiaAib = pabAibAia, (81)AiaAja = qijAjaAia, (82)AibAja = qij

pabAjaAib, (83)

AiaAjb = pab

pijAjbAia + (pab − 1

qab

)AibAja. (84)

These relations are valid for i < j and a < b. The up-per indices i and j are used to represent the row numberwhereas the lower indices a and b are used to representcolumn number of the matrix elements. All indices cantake the values 1 to n.To adapt multiparametric GL(n) to the matrix A, we com-pared the relations (81) to (84) with the relations (18)to (23) and obtained that Eqs. (18) to (23) coincide withthe Eqs. (81) to (84) forpij = q−1

ij = { 1 i < j ≤ d or d < i < j ≤ 2d,q−2 i ≤ d and d < j ≤ 2d.

This concludes that the homogeneous part of matrix T isa quantum group which can be denoted by GLq2,q−2 (2d).4. ConclusionIn this work, we have shown that the algebraic structuredefined by Eqs. (4) to (6) has an inhomogeneous invariancequantum group. Since the commutation relations satisfiedby the homogeneous part of this group are the same asthose commutation relations for GLq2,q−2 (2d), it is possi-ble to call the inhomogeneous invariance quantum groupof this system as BIGLq2,q−2 (2d), meaning the Bosonic

823

Author copy

The multi-dimensional q-deformed bosonic Newton oscillator and its inhomogeneous quantum invariance group

Inhomogeneous two parameter deformed General Linearquantum group. Thus, we have found a new property ofthe multidimensional deformed bosonic Newton oscilla-tor algebra. It is known that the homogeneous symmetrygroup of this algebra is SU(d) which can be found bytransforming ai’s to the set of {ai}. In this study, we con-sider rotation in the set of {ai, a∗i } and we show that it ispossible to obtain the inhomogeneous invariance quantumgroup for this system whose q-deformed characteristic isbased on the homogeneous part of this group.It is obvious, that within the limits q and H → 1, bosonicNewton oscillator algebra becomes usual boson algebra.H → 1 requires that χ3 = 1 and χ4 = 0 in the transfor-mation matrix T . Thus, in the limit q → 1, the algebrabecomes boson algebra. The transformation matrix, T ,belongs to the well known Bosonic Inhomogeneous Sym-plectic group (BISp), which is also a quantum group [12].Taking the inhomogeneous part as zero, one can see thatthe homogeneous part of the transformation matrix, T , be-comes Sp(2d, R).Besides the above limiting cases, one can also verify thephysical relevance of this study, if it can be applied to theq-deformed quantum field theory. The theory of quantumgroups can be helpful in the generalization of quantiza-tion, and obtain a more consistent approach to interactingfield theories.References

[1] L. D. Faddeev, N. Y. Reshetikhin, L. A. Takhtajan,Leningrad Mathematical Journal 1, 193 (1990)[2] M. Jimbo, Lett. Math. Phys. 11, 247 (1986)[3] V. G. Drinfeld, In: A. M. Gleason (Ed.), ProceedingsInternatioanl Congress of Mathematicians, 03-11 Au-gust 1986, Berkeley, California, USA (Amer. Math.Soc., 1988) 798[4] S. L. Woronowicz, Commun. Math. Phys. 111, 613(1987)[5] M. Chaichian, A. P. Demichev, Introducion to QuantumGroups (World Scientific River Edge, NJ, 1996)[6] Yu I. Manin, Quantum Groups and Non-commutativeGeometry (Centre de Reserches Mathematiques,Montreal, 1988)[7] M. Arik, A. Yildiz, J.Phys. A 30, L255 (1997)[8] M. Arik, M. A. Karaca, In: A. N. Sissakian (Ed.), Proc-cedings of the XII International Conference on Sym-metry Methods in Physics, 10-16 July 1993, Dubna,Russia (ICSMP 1995) 30[9] M. Arik, N. M. Atakishiyev, K. B. Wolf, J. Phys. A 32,L371 (1999)[10] A. Algin, M. Arik, N. M. Atakishiyev, Mod. Phys. Lett.

A 15, 1237 (2000)[11] M. Arik, U. Kayserilioglu, Balkan Phys. Lett 13, 101(2005)[12] M. Arik, A. Baykal, J. Math. Phys. 45, 4207 (2004)[13] A. A. Altintas, Czech. J. Phys. 56, 1069 (2006)[14] A. A. Altintas, M. Arik, Cent. Eur. J. Phys. 5, 70 (2007)[15] M. Arik, S. Gun, A. Yildiz, Eur. Phys. J. C 27, 453(2003)[16] A. A. Altintas, M. Arik, A. S. Arikan, Mod. Phys. Lett.A (in press)[17] A. Schirrmacher, J. Wess, B. Zumino, Z. Phys. C 49,317 (1991)[18] L. A. Takhtajan, Adv. Stu. P. M. 19, 1 (1989)[19] A. Schirrmacher, Z. Phys. C 50, 321 (1991)

824

Author copy