To give the complete list of Uq (sl2)-actions of the quantum plane, we first obtain the structure of quantum plane automorphisms. Then we introduce some special symbolic matrices to classify the series of actions using the weights. There are uncountably many isomorphism classes of the symmetries. We give the classical limit of the above actions

Duplij, S.

Sinel’shchikov, S.

English

CTU Open Journal Systems (Czech Technical University, Prague / České vysoké učení technické v Praze)

Acta Polytechnica Vol. 50 No. 5/2010On Uq (sl2)-actions on the Quantum PlaneS. Duplij, S. Sinel’shchikovAbstractTo give the complete list of Uq (sl2)-actions of the quantum plane, we ﬁrst obtain the structure of quantum planeautomorphisms. Then we introduce some special symbolic matrices to classify the series of actions using the weights.There are uncountably many isomorphism classes of the symmetries. We give the classical limit of the above actions.Keywords: quantum universal enveloping algebra, Hopf algebra, Verma module, representation, composition series,projection, weight.We present and classify Uq (sl2)-actions on the quan-tum plane [1]. The general form of an automorphismof the quantum plane [5] allows us to use the no-tion of weight. To classify the actions we introducea pair of symbolic matrices, which label the presenceof nonzero weight vectors. Finally, we present theclassical limit of the obtained actions.The deﬁnitions of a Hopf algebraH andH-action,the quantum universal enveloping algebra Uq (sl2)(determined by its generators k, k−1, e, f), and othernotations can be found in [3]. The quantum plane isa unital algebra Cq[x, y] generated by x, y and therelation yx = qxy, and we assume that 0 < q < 1.The notation Cq[x, y]i for the i-th homogeneouscomponent of Cq[x, y], being the linear span of themonomials xmyn with m + n = i, is used. De-note by (p)i the i-th homogeneous component ofa polynomial p ∈ Cq[x, y], that is the projectionof p onto Cq[x, y]i parallel to the direct sum of allother homogeneous components of Cq[x, y]. Denoteby C[x] and C[y] the linear spans of {xn|n ≥ 0} and{yn|n ≥ 0}, respectively. The direct sum decompo-sitions Cq[x, y] = C[x] ⊕ yCq[x, y] = C[y]⊕ xCq[x, y]is obvious. Let (P )x be a projection of a polynomialP ∈ Cq[x, y] to C[x] parallel to yCq[x, y].Proposition 1 Let Ψ be an automorphism ofCq[x, y], then there exist nonzero constants α, β suchthat [5]Ψ:x → αx, y → βy. (1)For any Uq (sl2)-action on Cq[x, y], we associate a2× 3 matrix, to be referred to as a full action matrixMdef=∥∥∥∥∥∥∥∥k (x) k (y)e (x) e (y)f (x) f (y)∥∥∥∥∥∥∥∥ . (2)An extension of Uq (sl2)-action from the generatorsto Cq[x, y] is given by (ab)udef= a (bu) , a (uv)def=Σi (a′iu) · (a′′i v) , a, b ∈ Uq (sl2) , u, v ∈ Cq[x, y] to-gether with the natural compatibility conditions [3].We have from (1) that the action of k is deter-mined by its action Ψ on x and y given by a 1 × 2matrix MkMkdef= ‖k (x) , k (y)‖ = ‖αx, βy‖ , (3)where α, β ∈ C\ {0}. This allows us to introduce theweight of xnym ∈ Cq[x, y] as wt (xnym) = αnβm.Another submatrix of M isMefdef=∥∥∥∥∥ e (x) e (y)f (x) f (y)∥∥∥∥∥ . (4)We call Mk and Mef an action k-matrix and an ac-tion ef -matrix, respectively.Each entry of M is a weight vector (by (3) and(1)), and all the nonzero monomials which constitutea speciﬁc entry have the same weight. We use thenotationwt (M)def=⎛⎜⎜⎝wt (k (x)) wt (k (y))wt (e (x)) wt (e (y))wt (f (x)) wt (f (y))⎞⎟⎟⎠ (5)⎛⎜⎜⎝wt (x) wt (y)q2wt (x) q2wt (y)q−2wt (x) q−2wt (y)⎞⎟⎟⎠=⎛⎜⎜⎝α βq2α q2βq−2α q−2β⎞⎟⎟⎠,where the matrix relation  is treated as a set ofelementwise equalities if they are applicable, that is,when the corresponding entry ofM is nonzero (henceadmits a well-deﬁned weight).Denote by (M)i the i-th homogeneous componentofM which, if nonzero, admits a well-deﬁned weight.Introduce the constants a0, b0, c0, d0 ∈ C such thatthe zero degree component of the full action matrix25Acta Polytechnica Vol. 50 No. 5/2010is(M)0 =⎛⎜⎜⎝0 0a0 b0c0 d0⎞⎟⎟⎠0. (6)We keep the subscript 0 to the matrix in the r.h.s.to emphasize the origin of this matrix as the 0-thhomogeneous component of M . Weights of nonzeroprojections of (weight) entries of M should have thesame weight, thenwt ((M)0) ⎛⎜⎜⎝0 0q2α q2βq−2α q−2β⎞⎟⎟⎠0. (7)All the entries of (M)0 are constants (6), and sowt ((M)0) ⎛⎜⎜⎝0 01 11 1⎞⎟⎟⎠0. (8)Let us use (Mef )i to construct a symbolic ma-trix( Mef)iwhose entries are symbols 0 or  in sucha way: a nonzero entry of (Mef )i is replaced by ,while a zero entry is replaced by the symbol 0. For0-th components the speciﬁc relations involved in (7)imply that each column of( Mef)0should contain atleast one 0, therefore we have 9 possibilities.Apply e and f to yx = qxy using (3) to getye (x)− qβe (x) y = qxe (y)− αe (y)x, (9)f (x) y − q−1β−1yf (x) = (10)q−1f (y)x− α−1xf (y) .We project (9)-(10) to Cq[x, y]1 and obtaina0 (1− qβ) y = b0 (q − α) x, d0(1− qα−1)x =c0(q − β−1) y, which givesa0 (1− qβ) = b0 (q − α) = (11)d0(1− qα−1) = c0 (q − β−1) = 0.Due to (11), weight constants α and β are1) a0 = 0 =⇒ β = q−1, (12)2) b0 = 0 =⇒ α = q, (13)3) c0 = 0 =⇒ β = q−1, (14)4) d0 = 0 =⇒ α = q. (15)We compare this to (7)–(8) and deduce that thesymbolic matrices containing two ’s should be ex-cluded. Using (7) and (12)–(15) we conclude thatthe position of  in the remaining symbolic matricesdetermines the associated weight constants( 00 0)0=⇒ α = q−2, β = q−1, (16)(0 0 0)0=⇒ α = q, β = q−2, (17)(0 0 0)0=⇒ α = q2, β = q−1, (18)(0 00 )0=⇒ α = q, β = q2, (19)and the matrix(0 00 0)0does not determine anyweight constants.In the 1-st homogeneous component we havewt (e (x)) = q2wt (x) = wt (x) (because 0 < q < 1),which implies (e (x))1 = a1y, and similarly we obtain(Mef )1 =(a1y b1xc1y d1x)1, (20)where a1, b1, c1, d1 ∈ C. So we introduce a sym-bolic matrix( Mef)1as above. The relations be-tween weights similar to (7) givewt((Mef )1)(q2α q2βq−2α q−2β)1 (21)(β αβ α)1.As a consequence we have that every row and ev-ery column of( M ef)1should contain at least one 0.We project (9)-(10) to Cq[x, y]2 and geta1 (1− qβ) y2 = b1 (q − α) x2, (22)d1(1− qα−1)x2 = c1 (q − β−1) y2, (23)whence a1 (1− qβ) = b1 (q − α) = d1(1− qα−1) =c1(q − β−1) = 0. So we obtain1) a1 = 0 =⇒ β = q−1, (24)2) b1 = 0 =⇒ α = q, (25)3) c1 = 0 =⇒ β = q−1, (26)4) d1 = 0 =⇒ α = q. (27)The symbolic matrix( 00 )1can be dis-carded from the list of symbolic matrices with atleast one 0 at every row or column, because of (21),26Acta Polytechnica Vol. 50 No. 5/2010(24)–(27). For other symbolic matrices with theabove property we have( 00 0)1=⇒ α = q−3, β = q−1, (28)(0 0 0)1=⇒ α = q, β = q−1, (29)(0 0 0)1=⇒ α = q, β = q−1, (30)(0 00 )1=⇒ α = q, β = q3, (31)(0  0)1=⇒ α = q, β = q−1, (32)and the matrix(0 00 0)1does not determine theweight constants.Let us introduce a table of families of Uq (sl2)-actions, each family is labeled by two symbolicmatrices( Mef)0,( Mef)1, and we call it a[( M ef)0;( Mef)1]-series.Note that the series labeled with pairs of nonzerosymbolic matrices at both positions are empty, be-cause each such matrix determines a pair of speciﬁcweight constants α and β (16)–(19) which fails to co-incide to any pair of such constants associated to theset of nonzero symbolic matrices at the second posi-tion (28)–(32). Also the series with zero symbolic ma-trix at the ﬁrst position and symbolic matrices con-taining only one  at the second position are empty.In this way we get 24 “empty”[( Mef)0;( M ef)1]-series.Let us turn to “non-empty” series and begin withthe case in which the action ef -matrix is zero.Theorem 2 The[(0 00 0)0;(0 00 0)1]-seriesconsists of four Uq (sl2)-module algebra structures onthe quantum plane given byk (x) = ±x, k (y) = ±y, (33)e (x) = e (y) = f (x) = f (y) = 0, (34)which are pairwise non-isomorphic.The next theorem describes the well-known sym-metry [6, 7].Theorem 3 The[(0 00 0)0;(0  0)1]-seriesconsists of a one-parameter (τ ∈ C \ {0}) familyof Uq (sl2)-module algebra structures on the quantumplanek (x) = qx, k (y) = q−1y, (35)e (x) = 0, e (y) = τx, (36)f (x) = τ−1y, f (y) = 0. (37)All these structures are isomorphic, in particularto the action as above with τ = 1.The essential claim here which is not coveredby [6, 7], is that no higher (> 1) degree terms couldappear in the expressions for e(x), e(y), f(x), f(y)in (36) and (37). This can be proved by a rou-tine computation which relies upon our assumption0 < q < 1.Consider the symmetries whose symbolic matrix( M ef)0contains one .Theorem 4 The[(0 0 0)0;(0 00 0)1]-seriesconsists of a one-parameter (b0 ∈ C \ {0}) familyof Uq (sl2)-module algebra structures on the quantumplanek (x) = qx, k (y) = q−2y, (38)e (x) = 0, e (y) = b0, (39)f (x) = b−10 xy, f (y) = −qb−10 y2. (40)All these structures are isomorphic, in particularto the action as above with b0 = 1.Theorem 5 The[(0 0 0)0;(0 00 0)1]-seriesconsists of a one-parameter (c0 ∈ C \ {0}) familyof Uq (sl2)-module algebra structures on the quantumplanek (x) = q2x, k (y) = q−1y, (41)e (x) = −qc−10 x2, e (y) = c−10 xy, (42)f (x) = c0, f (y) = 0. (43)All these structures are isomorphic, in particularto the action as above with c0 = 1.Theorem 6 The[( 00 0)0;(0 00 0)1]-seriesconsists of a three-parameter (a0 ∈ C \ {0}, s, t ∈ C)family of Uq (sl2)-actions on the quantum planek (x) = q−2x, k (y) = q−1y, (44)e (x) = a0, e (y) = 0, (45)f (x) =−qa−10 x2 + ty4, f (y) = −qa−10 xy + sy3. (46)27Acta Polytechnica Vol. 50 No. 5/2010The generic domain {(a0, s, t) |s = 0, t = 0} withrespect to the parameters splits into uncountablymany disjoint subsets{(a0, s, t) |s = 0, t = 0, ϕ = const} ,where ϕ =ta0s2. Each of these subsets correspondsto an isomorphism class of Uq (sl2)-module algebrastructures. Additionally there exist three more iso-morphism classes which correspond to the subsets{(a0, s, t) |s = 0, t = 0} ,{(a0, s, t) |s = 0, t = 0} , (47){(a0, s, t) |s = 0, t = 0} .The speciﬁc form of weights for x and y discardsprimordially all but ﬁnitely many terms (monomials)that could appear in the expressions for e(x), e(y),f(x), f(y) in (45) and (46). Thus it becomes mucheasier to establish the latter relations than to do thisfor the corresponding relations in the previous theo-rems. To prove the rest of the claims, one needs toguess the explicit form of the required isomorphisms.Theorem 7 The[(0 00 )0;(0 00 0)1]-seriesconsists of a three-parameter (d0 ∈ C \ {0}, s, t ∈ C)family of Uq (sl2)-actions on the quantum planek (x) = qx, k (y) = q2y, (48)e (x) =−qd−10 xy + sx3, e (y) = −qd−10 y2 + tx4, (49)f (x) = 0, f (y) = d0, (50)Here we have the domain {(d0, s, t) |s = 0, t = 0}which splits into the disjoint subsets {(d0, s, t) |s = 0 ,t = 0, ϕ = const} with ϕ = td0s2. This uncount-able family of subsets is in one-to-one correspon-dence to isomorphism classes of Uq (sl2)-module alge-bra structures. In addition, one also has three moreisomorphism classes which are labelled by the sub-sets {(d0, s, t) |s = 0, t = 0}, {(d0, s, t) |s = 0, t = 0},{(d0, s, t) |s = 0, t = 0}.Remark 8 The Uq (sl2)-symmetries on Cq[x, y] pic-ked from diﬀerent series are nonisomorphic, and theactions of k in diﬀerent series are diﬀerent.Remark 9 There are no Uq (sl2)-symmetries onCq[x, y] other than those presented in the above theo-rems, because the assumptions exhaust all admissibleforms for the components (Mef )0, (Mef )1 of the ac-tion ef -matrix.The associated classical limit actions of the Lie alge-bra sl2 (here it is the Lie algebra generated by e, f ,h subject to the relations [h, e] = 2e, [h, f ] = −2f ,[e, f ] = h) on C[x, y] by diﬀerentiations is derivedfrom the quantum action via substituting k = qh withsubsequent formal passage to the limit as q → 1.We present all quantum and classical actions inTable 1. Note that there exist more sl2-actions onC[x, y] by diﬀerentiations (see, e.g. [8]) than one cansee in Table 1. It follows from our results that therest of the classical actions admit no quantum coun-terparts. On the other hand, among the quantumactions listed in the ﬁrst row of Table 1, the onlyone to which the above classical limit procedure isapplicable, is the action with k(x) = x, k(y) = y.The remaining three actions of this series admit noclassical limit in the above sense.AcknowledgementOne of the authors (S.D.) is grateful to Yu. Be-spalov, J. Cuntz, B. Dragovich, J. Fuchs, A. Gavri-lik, H. Grosse, D. Gurevich, J. Lukierski, M. Pavlov,H. Steinacker, Z. Rakic´, W. Werner, and S. Woronow-icz for many fruitful discussions. Also, he would liketo thank M. Znojil for his invitation to the conference“Analytic and algebraic methods VI”, Villa Lanna,Prague, and kind hospitality at the Doppler Institutein Rez.References[1] Manin, Y. I.: Topics in Noncommutative Dif-ferential Geometry, Princeton University Press,Princeton, 1991.[2] Castellani, L., Wess, J. (eds.): Quantum groupsand their applications in physics, IOS Press, Am-sterdam, 1996.[3] Kassel, C.: Quantum Groups, Springer-Verlag,New York, 1995.[4] Sweedler, M. E.: Hopf Algebras, Benjamin, NewYork, 1969.[5] Alev, J., Chamarie, M.: De´rivations et automor-phismes de quelques alge`bres quantiques, Comm.Algebra 20, 1787–1802 (1992).[6] Montgomery, S., Smith, S. P.: Skew derivationsand Uq(sl(2)), Israel J. Math. 72, 158–166 (1990).[7] Lambe, L. A., Radford, D. E.: Introduction tothe Quantum Yang-Baxter Equation and Quan-tum Groups: An Algebraic Approach, Kluwer,Dordrecht, 1997.[8] Gonza´lez-Lo´pez, A., Kamran, N., Olver, P.:Quasi-exactly solvable Lie algebras of diﬀerentialoperators in two complex variables, J. Phys. A:Math. Gen. 24, 3 995–4 078 (1991).28Acta Polytechnica Vol. 50 No. 5/2010Table 1:Symbolic matricesUq-modulealgebrastructuresClassical limitsl2-actions bydiﬀerentiations[(0 00 0)0;(0 00 0)1] k (x) = ±x,k (y) = ±y,e (x) = e (y) = 0,f (x) = f (y) = 0,h (x) = 0, h (y) = 0,e (x) = e (y) = 0,f (x) = f (y) = 0,[(0 0 0)0;(0 00 0)1] k (x) = qx,k (y) = q−2y,e (x) = 0, e (y) = b0,f (x) = b−10 xy,f (y) = −qb−10 y2h (x) = x,h (y) = −2y,e (x) = 0, e (y) = b0,f (x) = b−10 xy,f (y) = −b−10 y2[(0 0 0)0;(0 00 0)1] k (x) = q2x,k (y) = q−1y,e (x) = −qc−10 x2,e (y) = c−10 xy,f (x) = c0, f (y) = 0,h (x) = 2x,h (y) = −ye (x) = −c−10 x2,e (y) = c−10 xy,f (x) = c0, f (y) = 0.[( 00 0)0;(0 00 0)1] k (x) = q−2x,k (y) = q−1y,e (x) = a0, e (y) = 0,f (x) = −qa−10 x2 + ty4,f (y) = −qa−10 xy + sy3.h (x) = −2x,h (y) = −y,e (x) = a0, e (y) = 0,f (x) = −a−10 x2 + ty4,f (y) = −a−10 xy + sy3.[(0 00 )0;(0 00 0)1] k (x) = qx, k (y) = q2y,e (x) = −qd−10 xy + sx3,e (y) = −qd−10 y2 + tx4,f (x) = 0,f (y) = d0,h (x) = x, h (y) = 2y,e (x) = −d−10 xy + sx3,e (y) = −d−10 y2 + tx4,f (x) = 0,f (y) = d0,[(0 00 0)0;(0  0)1] k (x) = qx,k (y) = q−1y,e (x) = 0, e (y) = τx,f (x) = τ−1y,f (y) = 0,h (x) = x, h (y) = −y,e (x) = 0,e (y) = τx,f (x) = τ−1y,f (y) = 0.Steven Duplij, Dr. habil.E-mail: steven.a.duplij@univer.kharkov.uahttp://webusers.physics.umn.edu/˜duplijTheory Group, Nuclear Physics LaboratoryV. N. Karazin Kharkov National University4 Svoboda Sq., 61077 Kharkov, UkraineSergey Sinel’shchikovE-mail: sinelshchikov@ilt.kharkov.uaMathematics DivisionB. I. Verkin Institute for Low Temperature Physicsand Engineering,47 Lenin Ave., Ukraine, 61103 Kharkov, Ukraine29

On Uq (sl2)-actions on the Quantum Plane

To give the complete list of Uq (sl2)-actions of the quantum plane, we first obtain the structure of quantum plane automorphisms. Then we introduce some special symbolic matrices to classify the series of actions using the weights. There are uncountably many isomorphism classes of the symmetries. We give the classical limit of the above actions.</p

On Uq (sl2)-actions on the Quantum Plane

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