Linear operators with invariant polynomial space and graded algebraY. Brihaye, Stefan Giller, and Piotr Kosinski Citation: Journal of Mathematical Physics 38, 989 (1997); doi: 10.1063/1.531806 View online: http://dx.doi.org/10.1063/1.531806 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/38/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Heisenberg algebra, umbral calculus and orthogonal polynomials J. Math. Phys. 49, 053509 (2008); 10.1063/1.2909731 Deficiency indices of operator polynomials in creation and annihilation operators J. Math. Phys. 44, 6209 (2003); 10.1063/1.1621060 The q-Laplace operator and q-harmonic polynomials on the quantum vector space J. Math. Phys. 42, 1326 (2001); 10.1063/1.1343092 Twisting invariance of link polynomials derived from ribbon quasi-Hopf algebras J. Math. Phys. 41, 5020 (2000); 10.1063/1.533390 Intertwining operators for a degenerate double affine Hecke algebra and multivariable orthogonal polynomials J. Math. Phys. 39, 4993 (1998); 10.1063/1.532505

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Linear operators with invariant polynomial spaceand graded algebra

Y. BrihayeDepartment of Mathematical Physics, University of Mons, Av. Maistriau,B-7000 MONS, Belgium

Stefan Giller and Piotr KosinskiDepartment of Theoretical Physics, University of Lodz, Pomorska 149/153,90-236 Lodz, Poland

~Received 14 May 1996; accepted for publication 26 July 1996!

The irreducible, finite-dimensional representations of the graded algebrasosp(j ,2) (j51,2,3) are expressed in terms of differential operators. Some quan-tum deformations of these algebras are shown to admit similar kinds of represen-tations. These are formulated in terms of finite difference operators. The results arediscussed in the framework of the quasi-exactly solvable equations. ©1997American Institute of Physics.@S0022-2488~97!00801-3#

I. INTRODUCTION

One attractive feature of quasi-exactly solvable~QES! equations1,2 is that they provide a niceinterplay between some spectral problems and some abstract algebraic structures. Following theapproach of Refs. 1, 3, and 4 it appears that the basic ingredients needed for the construction~andthe classification4! of QES equations are finite-dimensional representations of some algebra for-mulated in terms of differential operators of one or several variables, the so-called projectivizedrepresentations.

The most celebrated example is the algebra sl2 represented by the operators

J1~n,x!5x2d

dx2nx, J0~n,x!5x

d

dx2n

2, J2~n,x!5

d

dx. ~1!

If n is a positive integer, they preserve the space, sayP(n), of polynomials of degree at mostn inthe variablex. The linear differential operators possessing this property can be obtained as theelements of the enveloping algebra constructed overJ0 andJ6 .4

By performing an arbitrary change of variable and~or! a change of basis on these operators,one generates a large class of operators which possess a finite-dimensional invariant subspace. Asa consequence, a finite number of their eigenvectors can be found by solving an algebraic equa-tion. In this respect, these operators are called quasi-exactly solvable~see Ref. 5 for a recentreview!.

In this paper we address the classification of the operators that preserve the vector spaces ofcouples of polynomials with fixed degrees in one and in two variables. Sets of such operatorswhich generate all the others by means of polynomial combinations are exhibited. The~anti-!commutation relations between these basic elements are computed; as a general rule, they obey theaxioms of graded algebras. In the cases when the algebraic structures correspond to graded Liealgebras, we further consider quantum deformations of them and we construct representationsformulated in terms of finite difference operators. These representations are relevant for the un-derstanding of the discrete counterparts of quasi-exactly solvable equations.

0022-2488/97/38(2)/989/11/$10.00989J. Math. Phys. 38 (2), February 1997 © 1997 American Institute of Physics

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II. THE SCALAR QES OPERATORS

We briefly discuss this well-known case for completeness and for fixing some notations. Thealgebra sl2 is generated by three generators, sayH,X6 obeying

@H,X6#56X6 ,@X1 ,X2#52H. ~2!

The Casimir operator isC25H21$X2 ,X1%/2. The projectivized representation of dimensionn11 is constructed from the operators~1!:

H5J0~n,x!, X657J6~n,x!, C25n~n12!

4. ~3!

The SU~2! generators, sayJk(k51,2,3), are recovered by the combinations

J151

2~J22J1!, J25

i

2~J21J1!, J35J0 ~4!

~for shortness, we drop the dependence onn andx). The tensorial operators of the algebra can alsobe formulated in terms of the operatorsx andd/dx ~see the Appendix!.

Several deformations of the algebra sl2 are available.6 One of them is formulated by replacing

the commutators~2! by appropriateq-commutators:

q j0 j22 j2 j 052 j2 , q2 j1 j22 j2 j152~q11! j 0 , j 0 j12q j1 j 05 j1 , ~5!

whereq parametrizes the deformation. In Ref. 6 it is referred to as the second Witten’s deforma-tion. Standard formulas allow one to transformj into new operators whose~normal! commutatorsclose within the enveloping algebraUqsl2.

The counterpart of the operators~1! for the j was obtained in Ref. 4:

j15q2n/2~x2Dq2@n#qx!, j25q2n/2Dq , j 05q2n

q11

@2n12#q@n11#q

S xDq2@n#q@n11#q

@2n12#qD , ~6!

with the quantum symbol@ #q and the finite difference operatorDq defined by

@n#q[12qn

12q, Dqf[

f ~x!2 f ~qx!

~12q!x. ~7!

Obviously, the three operatorsj coincide with theJ’s of ~1! in the limit q→1.

III. THE 232 MATRIX QES OPERATORS

Let us considerP(m,n), the vector space of couples of polynomials of degreem(resp. n) inthe variablex for the first ~resp. second! component. Without losing generality, we assumem<n and we noteD[n2m. The linear differential operators preservingP(m,n) are the elementsof the enveloping algebra generated by the following 232 matrix operators:7

Te~n,D,x!5diag„Je~m,x!,Je~n,x!…, e50,6, ~8!

J~n,D!5 12diag~D1n,n!, ~9!

Qa~x!5qa~x!s2 , a51, . . . ,D11, ~10!

Qa~n,D,x!5qa~n,D,x!s1 , a51, . . . ,D11, ~11!

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with

qa~x!5xa21, ~12!

qa~n,D,x!5S )j50

D2a S x d

dx2~n112D!2 j D D S d

dxD a21

. ~13!

The operators above correspond respectively to the tensorial operatorspa of ~A6! andpD122a of~A7!. For brevity, we will drop~when the notation is not ambiguous! their dependence onD andx.

Many properties of the operators~8!–~11! are known.7 The operatorsTe form an sl2 algebra;theQa and theQD122a transform according to the representation of spins5D/2 under the adjointaction of theTe . The anticommutators$Qa ,Qb% can be expressed as polynomials of degreeD ofthe operators~8! and ~9!. In the caseD51 we have

$Q32a ,Qb%5~s3sas1!abTa1~ is2!ab J, a,b51,2, ~14!

wheresa(a51,2,3) are the Pauli matrices. In the caseD52 we obtain

$u i ,r j%51

2S $Ti ,Tj%22

3d i j C2D2 i e i jk S J2

1

2DTk2d i j SC2

61J~J21!

2 D , ~15!

where, for convenience, we introduced the following combinations:

u15Q32Q1

2, u25

Q31Q1

2i, u352Q2 , ~16!

r15Q12Q3

2, r25

Q11Q3

2i, r352Q2 . ~17!

The SU~2! generatorsTj ( j51,2,3) are defined from theTe(e50,6) as in ~4!; C2 in Eq. ~15!denotes the Casimir operator associated with the sl2 operators~8!, i.e.,

C2514diag„m~m12!,n~n12!… ~18!

~rememberm5n22 in this case!.Finally, let us notice that the operators

Te , J, Ka[mQa1nQD122a ~a51, . . . ,D11! ~19!

close under appropriate~anti-! commutator.

IV. THE CASE OSP(2,2)

A. The algebra

In the caseD51, the eight operators~8!–~11! form a representation of the graded Lie algebraosp~2,2!.3 Let us present it for completeness: it consists of four bosonic generators, denoted hereH, X6 , T, and of four fermionic ones, denotedV6 andV6 ~we use the notations of Ref. 8 apartfrom their operatorsJ6 which are noted hereX6). The ~anti-! commutation rules read

@H,X6#56X6 , @X6 ,T#50, @H,T#50, ~20!

@X1 ,X2#52H, ~21!

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@H6 12T,V6#50, @H7 1

2T,V6#56V6 , ~22!

@H6 12T,V6#56V6 , @H7 1

2T,V6#50, ~23!

$Vi ,Vj%5$Vi ,Vj%50 ~ i , j56 !, ~24!

$V6 ,V6%56 12X6 , ~25!

$V1 ,V2%52 12~H1 1

2T!, $V1 ,V2%52 12~H2 1

2T!. ~26!

The commutators between the operatorsX andV ~or X and V) can be deduced from~25! com-bined with the Jacobi identities:

@X6 ,V6#50, @X6 ,V6#50, ~27!

@X6 ,V7#5V6 , @X6 ,V7#5V6 . ~28!

The algebra has two Casimir operators.9 The quadratic one reads

C225H21X2X12 14T

222~V2V11V2V1!. ~29!

B. The representations

The operators~8!–~11! provide a family of representations of osp~2,2! by means of thefollowing identifications:

H5T0 , X657T6 , T52J, ~30!

V251

A2Q15

1

A2s2 , V15

1

A2Q25

1

A2xs2 , ~31!

V2521

A2Q25

21

A2d

dxs1 , V15

21

A2Q15

21

A2S x d

dx2nDs1 . ~32!

Given the integern the vector spaceP(n21,n) is preserved by~30!–~32!. Accordingly therepresentation has dimension 2n11. The corresponding value of the Casimir~29! is zero.

The generic, finite-dimensional, and irreducible representation of osp~2,2! ~Ref. 9! can as wellbe formulated in terms of differential operators. The space of the representation is the vector spaceP(n,n11,n21,n) ~i.e., the set of four-tuples of polynomials with degreen, n21, n11, n inone variable, sayx). The bosonic generators are of the form

H5diag„J0~n!,J0~n11!,J0~n21!,J0~n!…, ~33!

X657diag„J6~n!,J6~n11!,J6~n21!,J6~n!…, ~34!

T5diag~ t,t21,t21,t22!, ~35!

wheret is an arbitrary complex number.Owing thatV1 ,V2 andV1 ,V2 transform as two doublets under the sl2 subalgebra, the form

of these operators can be guessed in terms of appropriate tensorial operators:

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V2,151

A2 S 0 0 0 0

a21q1,2 0 0 0

a31q2,1~n! 0 0 0

0 a42q2,1~n11! a43q1,2 0

D , ~36!

V2,1521

A2 S 0 a12q2,1~n11! a13q1,2 0

0 0 0 a24q1,2

0 0 0 a34q2,1~n!

0 0 0 0

D . ~37!

The constantsai j have to be determined in such a way that the relations~24!–~26! are obeyed;these imply the following equations:

a12a241a13a3450, a42a211a43a3150,

a31a121a34a4250, a21a131a24a4350,

a34a435a12a21, a24a425a13a31,

a12a211a13a3151, a12a215t1n

2~n11!.

~38!

The most general solution9 depends on three free parameters~e.g.,a21,a31, anda43) which ~ifthey do not vanish! can be set to unity by a suitable similarity transformation. The representationis then fully specified by the values oft ~a complex number! and ofn ~an integer!:

a215a315a4352a4251, a125a34512a13511a245n1t

2~n11!. ~388!

The value of~29! is here nontrivial:C225(n/2)(n/211)2(t/2)(t/221) @the second Casimiroperator, sayC22, is equal to (t21)C22/2#. The atypical representation~30!–~32! corresponds tothe casea135a2450 which requirest5n12, leading to a null value for the two Casimir opera-tors.

Let us stress that the enveloping algebra constructed over the four operators~36! and ~37!generates the set of all QES operators preservingP(n,n11,n21,n) ~the general construction ofRef. 10 is bypassed in this case!. The underlying hidden symmetry of all QES systems constructedin this way is isomorphic to osp~2,2!.

C. Deformation of osp(2,2)

The general scheme for deformations of superalgebras is presented in Ref. 11. Here we willuse the deformation of osp~2,2! proposed by Deguchi.8 It can be reconciled with the pattern ofRef. 11 after a suitable redefinition of the generators. The deformed algebra is constructed in sucha way that the relations~24! and ~25! are kept, the latter defining the operatorsX6 . The defor-mation is then introduced through the anti-commutators~26! which become

$V1 ,V2%52 12@P1#q , $V1 ,V2%52 1

2@P2#q , ~39!

where

P6[H6 12T ~40!

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@Eq. ~39! above differs from Eq.~8! of Ref. 8 by a redefinition of the deformation parameterq→q2#.

The use of~25! and ~39! together with the Jacobi identities allows one to show that therelations~20!, ~22!, ~23!, and~27! are preserved while~21! and ~28! are deformed:

@X1 ,X2#5@P1#qqP21@P2#qq

P112S 121

qD ~V1V2qP11V1V2q

P2!, ~41!

@X6 ,V7#5V6qP721/261/2, ~42!

@X6 ,V7#5V6qP621/261/2. ~43!

In particular, the operatorsX6 ,H do not close~under commutator! within their enveloping algebrafor q Þ 1. As a consequence, the representation~6! cannot be used as a starting point for theconstruction of representations ofUq osp(2,2).

D. The representations

In order to construct the representations of interest for us, we assume~along with the unde-formed case! that the space of the representation is a direct sum of two subspaces respectivelyannihilated byV6 and V6 :

H5Hu%Hd , V6Hd50, V6Hu50, ~44!

and that the basic vectors are labelled according to their eigenvalues with respect to the commut-ing operatorsH andT:

Tuh,u&5tuuh,u&, Tuh,d&5tduh,d&, ~45!

Huh,u&5huh,u&, Huh,d&5huh,d&. ~46!

After some algebra, one can show that the representation is finite dimensional if

tu5n11, td5n, n integer, ~47!

2m

2<h<

m

2, 2

n

2<h<

n

2, n[m11, ~48!

with h and h varying by unit steps. It is worthwhile to identify the basic vectors with somemonomials in a suitable space of polynomials; let us pose

uh,d&[S 0

xh1n/2D , uh,u&5S xh1m/2

0D ~49!

Then, we further assume that the generatorsV andV have the following form, inspired fromEqs.~31! and ~32!:

V2521

A2s2 , V15

1

A2xs2 , ~50!

V2521

A2Dqs1 , V15

1

A2xn11Dqx

2ns1 . ~51!

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From the definition~25! one easily computes the action of the operatorsX6 on the basic vectors~49!:

X6uh,d&56F7n

21hG

q

uh61,d&, ~52!

X6uh,u&56F7m

21hG

q

uh61,u&. ~53!

This results in the following forms of the bosonic operators:

X252SDq 0

0 DqD , X15x2S xmDqx

2m 0

0 xnDqx2nD , ~54!

T5S n11 0

0 nD , qH5S 11~q21!x11m/2Dqx2m/2 0

0 11~q21!x11n/2Dqx2n/2D . ~55!

All generators ofUqosp(2,2) are then expressed in terms of the operatorDq and of the variablex. They preserveP(n21,n) and the representation has dimension 2n11.

The form of the operatorH is rather involved. In fact, we can insist from the beginning onhaving a Written type II deformed sl2 subalgebra, but then the form of the fermionic operators isuntractable. The reason we choose Deguchi’s deformation is that it allows for the particularlytransparent form~50! and ~51! for the fermionic operators.

V. THE CASE osp(1,2)

A. The algebra

In the caseD51 and ifmn Þ 0 in ~19!, the subalgebra generated by these operators coincideswith the graded Lie algebra osp~1,2!. This algebra plays a special role among graded Lie algebrassince it constitutes the simplest example of them, i.e., like SU~2! for simple Lie algebras. It hasfive generators: three bosonic ones,H,X6 , and two fermionic ones, sayv6 . The bosonic opera-tors form an sl2 subalgebra. The products involving the fermionic generators read

@H,v6#56 12v6 , @X6 ,v7#5v6 , @X6 ,v6#50, ~56!

$v1 ,v2%52 12H, $v6 ,v6%56 1

2X6 . ~57!

The Casimir operator is given by

C125H21 12$X1 ,X2%1@v1 ,v2#. ~58!

The projectivized representations of osp~1,2! are specified by the following identification using theoperators~8!–~11!:

H5T0 , X656T6 , ~59!

v15 12~Q11Q2!, v252 1

2~Q11Q2!. ~60!

Notice that these representations are equivalent to the generic, finite-dimensional ones; they leadto C125n(n11)/4.

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B. Deformation of osp(1,2)

We adopt the deformation of osp~1,2! which was elaborated in Ref. 12. It is generated by thethree elementsH,v6 together with the rules

@H,v6#56 12v6 , ~61!

$v1 ,v2%52 12@H#q . ~62!

We assume again that the space of the representation is the direct sum of two subspaces anduse, for the fermionic generators, the ansatz

v151

2S 0 a~d!x11aDx2a

c~d!x 0 D , v2521

2S 0 c~d!xbDx2b

a~d! 0 D ~63!

where a and b are constants whilea,a,c, and c are some functions the dilatation operatord[xD. Unlike the case osp~2,2! @see~50! and ~51!#, these quantities cannot be chosen as con-stants in order for~62! to be fulfilled. After some algebra, one finds that the relevant combinationsare

A~y![a~@y#q!a~@y#q!, C~y![c~@y#q!c~@y21#q!, ~64!

and thatA(y) andC(y) are regular functions of their argument provideda1b5n. The form~63!further indicates that the representation preservesP(n21,n) only in the casea5n andb50. Therelation ~62! finally leads to the following solution for the functionsA andC:

A~y!5

2F14Gq

F12Gq

S Fy2n

211

4Gq

2Fa2n

211

4Gq

@y2a#qD , C~y!5

2F14Gq

F12Gq

S Fy2n

211

4Gq

2Fb2n

221

4Gq

@y2b#qD .

~65!

These expressions are rather simple, however, the peculiar dependance of the functionsa, a, c, and c on @y#q ~rather than ony) in ~64! indicates that these quantities are ratherinvolved functions of the dilatation operatorxD.

VI. THE CASE osp(3,2)

Let us finally considerP„(m,k),(n,l )…, the vector space of couples of polynomials whose first~resp. second! component is of degreem (resp. n) in the variablex and of degreek(resp. l ) inthe variabley @notice thatP„(m,0),(n,0)…[P(m,n) previously defined#. For definiteness weassume thatm<n and k> l ~other cases can be treated very similarly! and we will use thenotationsD5n2m andD85k2 l . The linear differential operators preservingP„(m,k),(n,l )… arethe elements of the enveloping algebra generated by the following operators:

Se5diag„Je~n,x!,Je~m,x!…, e50,6, ~66!

Se85diag„Je~k,y!,Je~ l ,y!…, e50,6, ~67!

J5 12diag~D1n,n!, ~68!

J85 12diag~k,D81k!, ~69!

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Qa,a5qa~x!qa~y!s2, a51, . . . ,D11, a51, . . . ,D811, ~70!

Qa,a5qa~x!qa~y!s1, a51, . . . ,D11, a51, . . . ,D811 ~71!

~again we avoid writing the dependence of these operators on the dimensions!. The operatorsQa,a behave as a multiplet of spins5D/2 under the adjoint action of theSe and as a multiplet ofspin s85D8/2 with respect to the generationsSe8 . The same statement is true for theQa,as. Theanticommutators between these operators can be expressed in terms of the ones discussed in Sec.III:

$Qa,a ,Qb,b%5$Qa ,Qb%s1$Qb ,Qa%s1 . ~72!

These relations provide normal ordering rules on the set of operators preservingP„(m,k),(n,l )…. Owing ~72! and the results of Sec. III, one concludes that the operators underconsideration do not represent a Lie~super! algebra.

Let us further consider the linear combinations

Ka,a5Qa,D8122a1QD122a,a . ~73!

They form, together with the diagonal operators~66!–~69!, a subalgebra of the full structuregenerated by~66!–~71!. It is straightforward to evaluate the anticommutators

$Ka,a ,Kb,b%5$QD122a ,Qb%s1$QD8122b ,Qa%s11$QD122b ,Qa%s1$QD8122a ,Qb%s1 ,~74!

which, in general, are not linear combinations of the diagonal~bosonic! generators~66!–~69!.However, in the two cases

m50, n52, k51, l50 and m51, n53, k51, l50, ~75!

the anticommutators~74! reduce to a linear expressions in the diagonal operators. To be specificthey take the form (i , j ,k51,2,3, a,b51,2)

$Ki ,a ,Kj ,b%52m„2d i j ~ is2sk!abTk~y!1 i e i jk~ is2!abTk~x!…, ~76!

suggesting that the operators~73! constitute finite-dimensional representations of some graded Liealgebra. Indeed, it appears that the two sets of values~75! correspond to representations of thegraded algebra osp~3,2!. It is known13 that osp~3,2! admits only two representations of finitedimension~equal to 5 and 8!. All other representations are either infinite dimensional or plaguedwith a nonpositive definite metric of the space of the representation.

In order to obtain~76! we used~14! and ~15!, together with the following identities of theoperators~4!:

Jku~n50!P~0!50, $Jk ,Jt%u~n51!P~1!5 23C2dklP~1!. ~77!

They operate in such a way as to suppress the quadratic piece appearing on the right-hand side ofEq. ~15!.

VII. CONCLUSIONS

Recently, the generators~1!, ~5!, and~30!–~32! were used by A. Turbiner to solve a series ofgeneralizations of the Bochner problem.4,5,7 The original question is the following: classify thelinear differential operators, sayT, of k-order in one real variable such that the eigenvalue equa-tion Tf(x)5ef(x) has aninfinite sequence of orthogonal polynomial solutions.

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One can attenuate the hypothesis and pose the problem for afiniteset of polynomial solutions.This was solved in Ref. 4 for scalar equations of one variable.

One can also address Bochner’s problem for an enlarged number of variables or~and! forsystems of equations. For such extensions, it is crucial to classify the linear operators that preservetheN-dimensional vector space whose elements areN-tuples of polynomials of fixed degree. It issuch a classification that we presented here forN52, for one and for two real variables.

The operators constructed in the framework of deformations of the underlying algebras areuseful to address finite difference version of the Bochner problem.14,15

ACKNOWLEDGMENTS

We would like to thank Professor J. Lukierski for pointing Ref. 11 to our attention. S. G. andP. K. were supported by Grant No. KBN 2P 30221706 p 02.

APPENDIX: TENSORIAL OPERATORS

The tensorial operators of sl2 can be expressed in terms of the operatorsx andd/dx acting onan appropriate space of polynomials. Take the generatorsTe@e50,61, see Eq.~1!# and thetensorial operators of spins, sayPa(a51,•••,2s11), in the form

Te5diag„Je~m,x!,Je~n,x!…, Pa5paS x, ddxDs2 , ~A1!

wherem andn are two positive integers. Then the equations definingPa read

@Te,Pa#5„a212s~11e!…Pa1e . ~A2!

One finds the following solutions forpa :

paS x, ddxD5 (j50

a21

cjCa21j xa212 j S ddxD

s2 j1~m2n!/2

, ~A3!

whereCaj denotes the binomial coefficients while the parameterscjs obey the following recurrence

relation:

cj52cj21

~m2n12s1222 j !~m1n22s12 j !

4~2s2 j11!, 0< j<2s. ~A4!

The recurrence may be interrupted at any step, i.e., if

m5n22s, m5n22s12, . . . , m5n12s, ~A5!

leading to 2s11 possible values for the differencem2n and, correspondingly, to 2s11 sets ofoperatorsPa .

The operators corresponding tom5n22s andm5n12s are the relevant ones for the prob-lem treated in Sec. III. They can be rewritten as

pa5xa21, if m5n22s, ~A6!

pa5F )j50

a22 S x d

dx2n212 j D G S d

dxD 2s112a

, if m5n12s ~A7!

@p15(d/dx)2s is understood in the last formula#.

998 Brihaye, Giller, and Kosinski: Operators with invariant polynomial space

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999Brihaye, Giller, and Kosinski: Operators with invariant polynomial space

J. Math. Phys., Vol. 38, No. 2, February 1997

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