ON SOME OPERATOR IDENTITIES AND REPRESENTATIONS
1차 도함수의 연산자에 대해 일반화하는 것을 목표로 한다. 2변수 또는 k변수에 대한 연산자 표현이 가능하며, q-deformed 연산자의 경우도 가능하다.
Lie 대수로 연산자를 생성할 수 있으며, 이때 연산자는 n+1 차 도함수의 고유가치인 Pn(x)의 특성을 만족한다.
2변수에 대한 경우는 J1
2 (n) = x2∂x + xy∂y - nx로 연산자가 생성되며, n차 도함수의 연산자 ˜J1
2(n)도 생성된다.
k변수에 대한 경우도 마찬가지이며, q-deformed 연산자의 경우에는 q-다항식을 사용하여 연산자를 생성할 수 있다.
한글 요약 끝
ON SOME OPERATOR IDENTITIES AND REPRESENTATIONS
arXiv:funct-an/9305004v1 26 May 1993ON SOME OPERATOR IDENTITIES AND REPRESENTATIONSOF ALGEBRASALEXANDER TURBINER AND GERHARD POSTAbstract. Certain infinite families of operator identities related to powersof positive root generators of (super) Lie algebras of first-order differentialoperators and q-deformed algebras of first-order finite-difference operators arepresented.1.
The following operator identity holds(J+n )n+1 ≡(x2∂x −nx)n+1 = x2n+2∂n+1x, ∂x ≡ddx, n = 0, 1, 2, . .
. (1)The proof is straightforward:(i) the operator (J+n )n+1 annihilates the space of all polynomials of degree nothigher than n, Pn(x) = Span{xi : 0 ≤i ≤n};(ii) in general, an (n+ 1)−th order linear differential operator annihilating Pn(x)must have the form B(x)∂n+1x, where B(x) is an arbitrary function and(iii) since (J+n )n+1 is a graded operator, deg(J+n ) = +1, /footnoteso J+n maps xkto a multiple of xk+1 deg(J+nn+1) = n+1, hence B(x) = bx2n+2 while clearlythe constant b = 1.It is worth noting that taking the degree in (1) different from (n + 1), the l.h.s.
in(1) will contain all derivative terms from zero up to (n + 1)-th order.The identity has a Lie-algebraic interpretation. The operator (J+n ) is the positive-root generator of the algebra sl2 of first-order differential operators (the other sl2-generators are J0n = x∂x −n/2 , J−n = ∂x).
Correspondingly, the space Pn(x) isnothing but the (n + 1)-dimensional irreducible representation of sl2. The identity(1) is a consequence of the fact that (J+n )n+1 = 0 in this representation.There exist other algebras of differential or finite-difference operators (in morethan one variable), which admit a finite-dimensional representation.
This leads tomore general and remarkable operator identities. In the present Note, we showthat (1) is one representative of infinite family of identities for differential andfinite-difference operators.2.
The Lie-algebraic interpretation presented above allows us to generalize (1) forthe case of differential operators of several variables, taking appropriate degrees ofthe highest-positive-root generators of (super) Lie algebras of first-order differentialoperators, possessing a finite-dimensional invariant sub-space (see e.g.[1]). First weconsider the case of sl3.There exists a representation of sl3(C) as differentialoperators on C2.
One of the generators isJ12(n) = x2∂x + xy∂y −nxThe space Pn(x, y) = Span{xiyj : 0 ≤i + j ≤n} is a finite-dimensional represen-tation for sl3, and due to the fact (J12 (n))n+1 = 0 on the space Pn(x, y), hence we1
2ALEXANDER TURBINER AND GERHARD POSThave(J12 (n))n+1 = (x2∂x + xy∂y −nx)n+1 =k=n+1Xk=0n + 1kx2n+2−kyk∂n+1−kx∂ky ,(2)This identity is valid for y ∈C (as described above), but also if y is a Grassmannvariable, i.e. y2 = 0 1.
In the last case, J12(n) is a generator of osp(2, 2), see [1].More general (using slk instead of sl3), the following operator identity holds(Jk−2k−1 (n))n+1 ≡(x1kXm=1(xm∂xm −n))n+1 =xn+11Xj1+j2+...+jk=n+1Cn+1j1,j2,... ,jkxj11 xj22 . .
. xjkk ∂j1x1∂j2x2 .
. .
∂jkxk ,(3)where Cn+1j1,j2,... ,jk are the generalized binomial (multinomial) coefficients. If x ∈Ck,then Jk−2k−1 (n) is a generator of the algebra slk(C) [1], while some of the variablesx′s are Grassmann ones, the operator Jk−2k−1 (n) is a generator of a certain super Liealgebra of first-order differential operators.
The operator in l.h.s. of (3) annihilatesthe linear space of polynomials Pn(x1, x2, .
. .
xk) = Span{xj11 xj22 . .
. xjkk : 0 ≤j1 +j2 + .
. .
+ jk ≤n}.3. The above-described family of operator identities can be generalized for thecase of finite-difference operators with the Jackson symbol, Dx (see e.g.
[2])Dxf(x) = f(x) −f(q2x)(1 −q2)x+ f(q2x)Dxinstead of the ordinary derivative. Here, q is an arbitrary complex number.
Thefollowing operator identity holds( ˜J+n )n+1 ≡(x2Dx −{n}x)n+1 = q2n(n+1)x2n+2Dn+1x, n = 0, 1, 2, . .
. (4)(cf.
(1)), where {n} =1−q2n1−q2is so-called q-number.The operator in the r.h.s.annihilates the space Pn(x). The proof is similar to the proof of the identity (1).From algebraic point of view the operator ˜J+n is the generator of a q-deformed al-gebrasl2(C)qoffirst-orderfinite-differenceoperatorsontheline:˜J0n =xD −ˆn,˜J−n =D, where ˆn ≡{n}{n+1}{2n+2}(see [3] and also [1]), obeyingthe commutation relationsq2˜j0˜j−−˜j−˜j0 = −˜j−q4˜j+˜j−−˜j−˜j+ = −(q2 + 1)˜j0(5)˜j0˜j+ −q2˜j+˜j0 = ˜j+(˜j’s are related with ˜J’s through some multiplicative factors).
The algebra (5) hasthe linear space Pn(x) as a finite-dimensional representation.An attempt to generalize (2) replacing continuous derivatives by Jackson symbolsimmediately leads to requirement tp introduce the quantum plane and q-differentialcalculus [4]xy = qyx ,1In this case just two terms in l.h.s. of (2) survive.
ON SOME OPERATOR IDENTITIES . .
.3Dxx = 1 + q2xDx + (q2 −1)yDy,Dxy = qyDx ,Dyx = qxDy,Dyy = 1 + q2yDy ,DxDy = q−1DyDx . (6)The formulae analogous to (2) have the form( ˜J12 (n))n+1 ≡(x2Dx + xyDy −{n}x)n+1 =k=n+1Xk=0q2n2−n(k−2)+k(k−1)n + 1kqx2n+2−kykDn+1−kxDky ,(7)wherenkq≡{n}!{k}!
{n −k}! , {n}!
= {1}{2} . .
. {n}are q-binomial coefficient and q-factorial, respectively.
Like all previous cases, ify ∈C, the operator ˜J12(n) is one of generators of q-deformed algebra sl3(C)q offinite-difference operators, acting on the quantum plane and having the linear spacePn(x, y) = Span{xiyj : 0 ≤i + j ≤n} as a finite-dimensional representation; thel.h.s. of (7) annihilates Pn(x, y).
If y is Grassmann variable, ˜J12(n) is a generator ofthe q-deformed superalgebra osp(2, 2)q possessing finite-dimensional representation.Introducing a quantum hyperplane [4], one can generalize the whole family ofthe operator identities (3) replacing continuous derivatives by finite-difference op-erators.One of us (A.T.) acknowledges of Profs. L. Michel, R. Thom and F. Pham forkind hospitality and their interest to the present work.
4ALEXANDER TURBINER AND GERHARD POSTReferences[1] A.V. Turbiner, “Lie algebras and linear operators with invariant subspace”, Preprint IHES-92/95 (1992); to appear in “Lie algebras, cohomologies and new findings in quantum mechan-ics”, Contemporary Mathematics, AMS, 1993, N. Kamran and P. Olver (eds.
)[2] H. Exton, “q-Hypergeometrical functions and applications”, Horwood Publishers, Chichester,1983. [3] O. Ogievetsky and A. Turbiner, “sl(2, R)q and quasi-exactly-solvable problems”, PreprintCERN-TH: 6212/91 (1991) (unpublished).
[4] J. Wess and B. Zumino, “Covariant differential calculus on the quantum hyperplane”, Nucl.Phys. B18 302 (1990)(Proc.
Suppl. ); B. Zumino, Mod.
Phys. Lett.
A6 1225 (1991)Institute for Theoretical and Experimental Physics, Moscow 117259, RussiaandDeparment of Applied Mathematics, University of Twente, P.O. Box 217,7553 CVEnschede, HollandE-mail address:TURBINER@CERNVM or TURBINER@VXCERN.CERN.CHand POST@MATH.UTWENTE.NLCurrent address: I.H.E.S., Bures-sur-Yvette, F-91440, France
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