String Equations of the q-KP Hierarchy

STRING EQUA TIONS OF THE Q-KP HIERARCHY KELEI TIAN † , JINGSONG HE ∗ † ‡ , YUCAI SU † AND YI CHENG † † Dep artment of Mathematics, Un iversity of Scienc e and T e c hnolo gy of China, H efei, 230026 Anhui, China ‡ Dep artment of Mathematics, Ningb o University, Ningb o, 31521 1 Z hejiang, China Abstract. Based on the Lax o per ator L a nd Orlov-Sh ulman’s M op erator , the string equa- tions of the q - K P hierarchy ar e establis hed from sp ecial additiona l s y mmetry flows, and the negative Virasoro cons traint generators { L − n , n ≥ 1 } of th e 2 − reduced q - KP hier arch y ar e also obtained. Keyw ords: q - KP hierarc hy , additio na l symmetry , string equations, Virasoro constrain t Mathematics Sub ject Classification(2000): 35Q53, 37K05, 37K10 P A CS(2003 ): 02.30.Ik 1. Introduction The q -deformed in tegrable system (also called q - analogue or q -deformation of classical inte- grable system ) is defined by means of q -deriv ativ e ∂ q [1, 2] instead of usual deriv ativ e ∂ with resp ect to x in a classical system. I t r educes to a classical in tegrable system as q → 1. Re- cen tly , the q -deformed Ka dom tsev-P etviash vili ( q -KP) hierarc h y is a sub ject of in tensiv e study in the literature from [3] to [14]. Its infinite conserv ation laws , bi-Hamiltonian structure, τ function, additional symmetries and its constrained sub-hierarc hy hav e already b een rep ort ed in [4, 5, 11, 12, 14]. The additional symmetries, string equations and Virasoro constraints of the KP hierarch y are imp ortant as they are in v olve d in the matrix mo dels of the string theory [15]. F or example, there are sev eral new w orks [16–2 0] on this topic. The additio na l symmetries w ere disco v ered indep enden tly at least t wice b y Sato Sc ho ol [21] and Orlo v-Shulman [22], in quite differen t en vironmen ts and philosoph y altho ug h they are equiv alen t ess entially . It is w ell-know n that L.A.Dic k ey [23] presen ted a v ery elegan t and compact pro of of Adler-Shiota-v an Mo erb ek e (ASvM) form ula [24, 2 5 ] b ased on the Lax op erator L and Orlov and Sh ulman’s M o p erator [22], and g av e t he string equation a nd the a ctio n of the additional symmetries o n the τ function of the classical KP hierarc h y . S.P anda and S.Roy gav e the Virasoro and W - constraints on the τ function of the p -reduced KP hierarc h y by expanding the additional symmetry o p erator in terms o f the Lax op erator [26, 27]. It is quite intere sting to study the analogous pro p erties of q -defo r med KP hierarc hy by this expanding metho d . The main purp ose of this article is to giv e the string equations of the q -KP hierarc h y , and t hen study the negative Virasoro constrain t generators { L − n , n ≥ 1 } of 2- reduced q -KP hierarc hy . The orga nization of this paper is as follo ws. W e recall some basic results and additional symmetries of q -KP hierarc hy in Section 2 . The string equations are given in Sections 3. The Virasoro constrain ts on the τ function of the 2-reduced ( q -K dV) hierarc hy are studied in Section 4. Section 5 is dev oted to conclusions and discussions. ∗ Corresp o nding author . Email: jshe@ustc.edu.cn, he jing s ong@nbu.edu.cn. 1 2 KELEI TIAN † , JIN GSONG H E ∗ † ‡ , YUCAI S U † AN D Y I CHENG † A t the end of the this section, we shall collect some useful fa cts of q -calculus [2] to mak e this pap er b e self-contained . The q -deriv ative ∂ q is defined b y ∂ q ( f ( x )) = f ( q x ) − f ( x ) x ( q − 1) (1.1) and the q -shift op erator is θ ( f ( x )) = f ( q x ) . (1.2) ∂ q ( f ( x )) reco ve rs the ordinary differen tiat ion ∂ x ( f ( x )) as q go es to 1. Let ∂ − 1 q denote the formal in v erse of ∂ q . In general the following q - deformed Leibnitz rule holds ∂ n q ◦ f = X k ≥ 0  n k  q θ n − k ( ∂ k q f ) ∂ n − k q , n ∈ Z (1.3) where the q -n umber and the q -binomial are defined by ( n ) q = q n − 1 q − 1 ,  n k  q = ( n ) q ( n − 1) q · · · ( n − k + 1) q (1) q (2) q · · · ( k ) q ,  n 0  q = 1 . F or a q -pseudo-differen tia l o p erator( q -PDO) of the form P = P n i = −∞ p i ∂ i q , w e separate P into the differen tial par t P + = P i ≥ 0 p i ∂ i q and the inte gr al part P − = P i ≤− 1 p i ∂ i q . The conjugate op eration “ ∗ ” for P is defined b y P ∗ = P i ( ∂ ∗ q ) i p i with ∂ ∗ q = − ∂ q θ − 1 = − 1 q ∂ 1 q , ( ∂ − 1 q ) ∗ = ( ∂ ∗ q ) − 1 = − θ ∂ − 1 q . The q -exp onen t e x q is defined as follo ws e x q = ∞ X n =0 x n ( n ) q ! , ( n ) q ! = ( n ) q ( n − 1) q ( n − 2) q · · · (1) q . Its equiv alen t expression is of the form e x q = exp ( ∞ X k =1 (1 − q ) k k (1 − q k ) x k ) , (1.4) whic h is crucial to dev elop the τ function of the q -KP hierarc h y [11]. 2. q -KP hierarchy and its additional s ymmetries Similar to the general w ay of describing the classical KP hierarc h y [21, 28], w e first give a brief intro duction of q - KP hierarc hy and its additio na l symmetries based on [1 1 , 12]. Let L b e o ne q -PDO give n b y L = ∂ q + u 0 + u − 1 ∂ − 1 q + u − 2 ∂ − 2 q + · · · , (2.1) whic h are called L a x op erator of q -KP hierarc h y . There exist infinite n um b er of q - partial dif- feren tial equations related to dynamical v ariables { u i ( x, t 1 , t 2 , t 3 , · · · , ) , i = 0 , − 1 , − 2 , − 3 , · · · } and can b e deduced from the generalized Lax equation, ∂ L ∂ t n = [ B n , L ] , n = 1 , 2 , 3 , · · · , (2.2) STRING EQU A TIONS OF THE Q- KP H IERARCHY 3 whic h are called q - KP hierarc h y . Here B n = ( L n ) + = n P i =0 b i ∂ i q and L n − = L n − L n + . L in eq. (2.1) can b e generated b y dressing op erator S = 1 + P ∞ k =1 s k ∂ − k q in the following w a y L = S ◦ ∂ q ◦ S − 1 . (2.3) Dressing op erator S satisfies Sato equation ∂ S ∂ t n = − ( L n ) − S, n = 1 , 2 , 3 , · · · . (2.4) The q -w av e function w q ( x, t ; z ) and the q -adjoint f unction w ∗ q ( x, t ; z ) are g iven b y w q = S e xz q exp( ∞ X i =1 t i z i ) , w ∗ q ( x, t ; z ) = ( S ∗ ) − 1 | x/q e − xz 1 /q exp( − ∞ X i =1 t i z i ) , whic h satisfies fo llo wing linear q -differential equations Lw q = z w q , L ∗ | x/q w ∗ q = z w ∗ q . Here the notation P | x/t = P i P i ( x/t ) t i ∂ i q is used for P = P i p i ( x ) ∂ i q . F urthermore, w q ( x, t ; z ) and w ∗ q ( x, t ; z ) can b e expressed b y sole function τ q ( x ; t ) [11] as w q = τ q ( x ; t − [ z − 1 ]) τ q ( x ; t ) e xz q exp ∞ X i =1 t i z i ! = e xz q e ξ ( t,z ) e − P ∞ i =1 z − i i ∂ i τ q τ q , (2.5) w ∗ q = τ q ( x ; t + [ z − 1 ]) τ q ( x ; t ) e − xz 1 /q exp − ∞ X i =1 t i z i ! = e − xz 1 /q e − ξ ( t,z ) e + P ∞ i =1 z − i i ∂ i τ q τ q , where [ z ] =  z , z 2 2 , z 3 3 , . . .  . The follow ing Lemma show s there exist an essen tial corresp ondence b etw een q -KP hierarc hy and KP hierarc h y . Lemma 1. [11] Let L 1 = ∂ + u − 1 ∂ − 1 + u − 2 ∂ − 2 + · · · , where ∂ = ∂ /∂ x , b e a solution of the classical KP hierarc hy and τ b e its tau f unction. Then τ q ( x, t ) = τ ( t + [ x ] q ) is a tau function of the q -K P hierarc h y asso ciated with Lax op erat o r L in eq. (2.1), where [ x ] q =  x, (1 − q ) 2 2(1 − q 2 ) x 2 , (1 − q ) 3 3(1 − q 3 ) x 3 , · · · , (1 − q ) i i (1 − q i ) x i , · · ·  . Define Γ q and Or lov-Sh ulman’s M o p erator Γ q = ∞ X i =1  it i + (1 − q ) i (1 − q i ) x i  ∂ i − 1 q , (2.6) 4 KELEI TIAN † , JIN GSONG H E ∗ † ‡ , YUCAI S U † AN D Y I CHENG † M = S Γ q S − 1 . (2.7) Dressing [ ∂ k − ∂ k q , Γ q ]=0 give s ∂ k M = [ B k , M ] . (2.8) Eq. (2.2) together with eq. (2.8) implies that ∂ k ( M m L n ) = [ B k , M m L n ] . (2.9) Define the additional flow s for each pair m, n a s follows ∂ S ∂ t ∗ m,n = − ( M m L n ) − S, (2.10) or equiv alen tly ∂ L ∂ t ∗ m,n = − [( M m L n ) − , L ] , (2.11) ∂ M ∂ t ∗ m,n = − [( M m L n ) − , M ] . (2.12) The additional flow s ∂ ∗ mn = ∂ ∂ t ∗ m,n comm ute with t he hierarc hy , i.e. [ ∂ ∗ mn , ∂ k ] = 0 but do not comm ute with each other, so they are additiona l symmetries [12]. ( M m L n ) − serv es as the generator of the additional symmetries along the tra j ectory parametrized by t ∗ m,n . 3. String equa tions of the q -KP hierarchy In this section w e shall get string equations for the q -KP hierarch y from sp ecial additional symmetry flows. F or this, w e need a lemma. Lemma 2. The f o llo wing equation [ M , L ] = − 1 (3.1) holds. Pro of. Direct calculations show that [Γ q , ∂ q ] = " ∞ X i =1  it i + (1 − q ) i 1 − q i x i  ∂ i − 1 q , ∂ q # = ∞ X i =1 h (1 − q ) i 1 − q i x i ∂ i − 1 q , ∂ q i = ∞ X i =1 (1 − q ) i 1 − q i  x i ∂ i q − ( ∂ q ◦ x i ) ∂ i − 1 q  = ∞ X i =1 (1 − q ) i 1 − q i  x i ∂ i q − (( ∂ q x i ) + q i x i ∂ q ) ∂ i − 1 q  = ∞ X i =1 (1 − q ) i 1 − q i  (1 − q i ) x i ∂ i q − 1 − q i 1 − q x i − 1 ∂ i − 1 q  STRING EQU A TIONS OF THE Q- KP H IERARCHY 5 = ∞ X i =1 ((1 − q ) i x i ∂ i q − (1 − q ) i − 1 x i − 1 ∂ i − 1 q ) = − 1 , where w e ha ve used [ t i , ∂ q ] = 0 in the second step and ∂ q ◦ x i = ( ∂ q x i ) + q i x i ∂ q in the fo urt h step. Then [ M , L ] = [ S Γ q S − 1 , S ∂ q S − 1 ] = S [Γ q , ∂ q ] S − 1 = − 1 .  By virtue of Lemma 2, w e ha v e Corollary 1. [ M , L ] = − 1 implies [ M , L n ] = − nL n − 1 . Therefore, [ M L − n +1 , L n ] = − n. (3.2) The a ctio n of additional flo ws ∂ ∗ 1 , − n +1 on L n are ∂ ∗ 1 , − n +1 L n = − [( M L − n +1 ) − , L n ], whic h can b e written as ∂ ∗ 1 , − n +1 L n = [( M L − n +1 ) + , L n ] + n. (3.3) The following theorem holds by virtue o f eq.(3.3). Theorem 1. If an op erator L do es no t depend on the parameters t n and the additional v ariables t ∗ 1 , − n +1 , then L n is a purely differen tial o p erator, and the string equations of the q -KP hierarc h y are give n by [ L n , 1 n ( M L − n +1 ) + ] = 1 , n = 2 , 3 , 4 , · · · (3.4) In view of the additional symmetries a nd string equations, w e can get the follow ing corollary , whic h plays a crucial role in the study of the constrain ts on the τ function of the p-reduced q -K P hierarc hy . Corollary 2. If L n is a differen tial op erator, and ∂ ∗ 1 , − n +1 S = 0, then ( M L − n +1 ) − = n − 1 2 L − n , n = 2 , 3 , 4 , · · · (3.5) Pro of. Since [ M , L ] = − 1, it is not difficult to obta in [ M , L − n +1 ] = ( n − 1) L − n , and hence ( M L − n +1 ) − − ( L − n +1 M ) − = ( n − 1) L − n . (3.6) Noticing [( n − 1) L − n , L n ] = 0, then [( M L − n +1 ) − − ( L − n +1 M ) − , L n ] = 0 , i.e. , [( M L − n +1 ) − , L n ] = [( L − n +1 M ) − , L n ] . Th us ∂ ∗ 1 , − n +1 L n = − [( L − n +1 M ) − , L n ] = − 1 2 [( M L − n +1 ) − + ( L − n +1 M ) − , L n ] , 6 KELEI TIAN † , JIN GSONG H E ∗ † ‡ , YUCAI S U † AN D Y I CHENG † or equiv alen tly ∂ ∗ 1 , − n +1 S = − 1 2 ( M L − n +1 + L − n +1 M ) − S. Therefore, it follows from the equation ∂ ∗ 1 , − n +1 S = 0 that ( M L − n +1 + L − n +1 M ) − = 0 . Com bining this with (3.6 ) finishes the pro of.  4. Constraints on t he τ function of the q -KdV hierarchy In this section, w e mainly study the a sso ciated constraints on τ - function of the 2-r educed q -K P ( q -KdV) hierar ch y from string equations eq. (3.4) . T o this end, w e first define residue res L = u − 1 of L giv en by eq. (2 .1 ) and state t w o v ery useful lemmas. Lemma 3. F or n = 1 , 2 , 3 , · · · , res L n = ∂ 2 log τ q ∂ t 1 ∂ t n . (4.1) where τ q is the tau function of the q -KP hierarch y . Pro of. T aking the residue of ∂ S ∂ t n = − ( L n ) − S , w e get ∂ s 1 ∂ t n = − res( ( L n ) − (1 + s 1 ∂ − 1 q + s 2 ∂ − 2 q + · · · )) = − res( L n ) − = − res L n . Noting that u 0 = s 1 − θ ( s 1 ) = − x ( q − 1) ∂ q s 1 = x ( q − 1) ∂ q ∂ t 1 log τ q , s 1 = − ∂ l og τ q ∂ t 1 (see [1 4]), then res L n = − ∂ s 1 ∂ t n = ∂ 2 log τ q ∂ t 1 ∂ t n .  Lemma 4. Orlov-Sh ulman’s M op erator ha s the expansion of the form M = ∞ X i =1  it i + (1 − q ) i (1 − q i ) x i  L i − 1 + ∞ X i =1 V i +1 L − i − 1 , (4.2) where V i +1 = − i X a 1 +2 a 2 +3 a 3 + ··· = i ( − 1) a 1 + a 2 + ··· ( ∂ t 1 ) a 1 a 1 ! ( 1 2 ∂ t 2 ) a 2 a 2 ! ( 1 3 ∂ t 3 ) a 3 a 3 ! · · · log τ q . Pro of. First, we assert M w q = ∂ w q ∂ z . Inde ed, fr o m the iden tity ∂ i − 1 q e xz q = z i − 1 e xz q w e hav e tha t M w q = S Γ q S − 1 S e xz q e ξ ( t,z ) = S ∞ X i =1  it i + (1 − q ) i 1 − q i x i  z i − 1 ! e xz q e ξ ( t,z ) , where ξ ( t, z ) = P ∞ i =1 t i z i . On the o t her hand, ∂ w q ∂ z = ∂ ( S e xz q e ξ ( t,z ) ) ∂ z = S  ∂ e xz q ∂ z e ξ ( t,z ) + e xz q ∂ e ξ ( t,z ) ∂ z  = S ∞ X i =1  it i + (1 − q ) i 1 − q i x i  z i − 1 ! e xz q e ξ ( t,z ) . STRING EQU A TIONS OF THE Q- KP H IERARCHY 7 Th us the assertion is v erified. Next, by a direct calculatio n fro m eq.(1.4) and eq.(2.5), w e ha ve log w q = ∞ X k =1 (1 − q ) k k (1 − q k ) ( xz ) k + ∞ X n =1 t n z n + ∞ X N =0 1 N ! ( − ∞ X i =1 z − i i ∂ i ) N log τ q − log τ q . (4.3) Let M = P ∞ n =1 a n L n − 1 + P ∞ n =1 b n L − n . Then in light of Lw q = z w q and t he assertion men t io ned in ab o ve , w e o bta in ∂ w q ∂ z = M w q = ( ∞ X n =1 a n L n − 1 + ∞ X n =1 b n L − n ) w q , and hence ∂ log w q ∂ z = 1 w q ∂ w q ∂ z = ∞ X n =1 a n z n − 1 + ∞ X n =1 b n z − n . (4.4) Th us by comparing the co efficien ts of z in ∂ l og w q ∂ z giv en by eq. (4 .3) and eq. (4.4), a i and b i are determined suc h tha t M is obtained as eq. (4.2) .  T o b e an in tuitiv e glance, the first few V i +1 are g iven as follows. V 2 = ∂ log τ q ∂ t 1 , V 3 = ∂ log τ q ∂ t 2 − ∂ 2 log τ q ∂ t 2 1 , V 4 = ( 1 2 ∂ 3 ∂ t 3 1 − 3 2 ∂ 2 ∂ t 1 ∂ t 2 + ∂ ∂ t 3 ) log τ q , V 5 = ( − 1 3! ∂ 4 ∂ t 4 1 − 1 2 ∂ 2 ∂ t 2 2 − 4 3 ∂ 2 ∂ t 1 ∂ t 3 + ∂ ∂ t 4 ) log τ q , V 6 = ( 1 4! ∂ 5 ∂ t 5 1 − 5 12 ∂ 4 ∂ t 3 1 ∂ t 3 + 5 6 ∂ 3 ∂ t 2 1 ∂ t 3 − 5 4 ∂ 2 ∂ t 1 ∂ t 4 − 5 6 ∂ 2 ∂ t 2 ∂ t 3 + ∂ ∂ t 5 ) log τ q . No w w e consider the 2-reduced q -KP hierarc hy( q - KdV hierarc h y), b y setting L 2 − = 0 or setting L 2 = ∂ 2 q + ( q − 1) xu∂ q + u. (4.5) T o mak e the fo llowing theorem b e a compact form, in tro duce L − n = 1 2 ∞ X i =2 n +1 i 6 =0(mod 2) i ˜ t i ∂ ∂ ˜ t i − 2 n + 1 4 X k + l = n +1 (2 k − 1)(2 l − 1 ) ˜ t 2 k − 1 ˜ t 2 k − 1 (4.6) and ˜ t i = t i + (1 − q ) i i (1 − q i ) x i , i = 1 , 2 , 3 , · · · . (4.7) Theorem 2. If L 2 satisfies eq. (3.4), the Virasoro constraints imp osed o n the τ -function o f the q -K dV hierarc hy are L − n τ q = 0 , n = 1 , 2 , 3 , · · · , (4.8) and the Virasoro comm utation relations [ L − n , L − m ] = ( − n + m ) L − ( n + m ) , m, n = 1 , 2 , 3 , · · · (4.9) hold. 8 KELEI TIAN † , JIN GSONG H E ∗ † ‡ , YUCAI S U † AN D Y I CHENG † Pro of. F or n = 1 , 2 , 3 , · · · , we ha v e res( M L − 2 n +1 ) = res( M L − 2 n +1 ) − = r es( − 2 n + 1 2 L − 2 n ) − = 0 (4.10) with the help of eq. (3.5). Substituting the expansion of M in eq. (4.2) into eq. (4.10), then ∞ X i =1  it i + (1 − q ) i 1 − q i x i  res L i − 2 n + ∞ X i =1 res( V i +1 L − i − 2 n ) = 0 , whic h implies ∞ X i =2 n +1 i 6 =0(mod 2)  it i + (1 − q ) i 1 − q i x i  res L i − 2 n + (2 n − 1) t 2 n − 1 + (1 − q ) 2 n − 1 1 − q 2 n − 1 x 2 n − 1 = 0 . (4.11) Substituting res L i − 2 n = ∂ 2 log τ q ∂ t 1 ∂ t i − 2 n in to eq. (4 .1 1), then p erforming an in tegra tion with resp ect to t 1 and multiplying by τ q 2 , it b ecomes ˜ L − n τ q = 0 , n = 1 , 2 , 3 , · · · , where ˜ L − n = 1 2 ∞ X i =2 n +1 i 6 =0(mod 2)  it i + (1 − q ) i 1 − q i x i  ∂ ∂ t i − 2 n + (1 − q ) 2 n − 1 1 − q 2 n − 1 · 1 2 t 1 x 2 n − 1 + 1 2 (2 n − 1) t 1 t 2 n − 1 + C ( t 2 , t 3 , · · · ; x ) . (4.12) The inte gra tion constant C ( t 2 , t 3 , · · · ; x ) with resp ect to t 1 could b e the arbitrary function with the parameters ( t 2 , t 3 , · · · ; x ). What w e will do is to determine C ( t 2 , t 3 , · · · ; x ) suc h that ˜ L − n satisfy Virasoro comm utatio n relatio ns. Let ˜ t i = t i + (1 − q ) i i (1 − q i ) x i , i = 1 , 2 , 3 , · · · , and choose C ( t 2 , t 3 , · · · ; x ) as C ( t 2 , t 3 , · · · ; x ) = − 1 4 2 n − 3 X k =3 (2 k − 1)(2 n − 2 k + 1)  t 2 k − 1 + (1 − q ) 2 k − 1 (2 k − 1)(1 − q 2 k − 1 ) x 2 k − 1  ·  t 2 n − 2 k + 1 + (1 − q ) 2 n − 2 k + 1 (2 n − 2 k + 1 ) ( 1 − q 2 n − 2 k + 1 ) x 2 n − 2 k + 1  − 1 2 (2 n − 1) x  t 2 n − 1 + (1 − q ) 2 n − 1 (2 n − 1)(1 − q 2 n − 1 ) x 2 n − 1  , Then ˜ L − n = 1 2 ∞ X i =2 n +1 i 6 =0(mod 2) i ˜ t i ∂ ∂ ˜ t i − 2 n + 1 4 X k + l = n +1 (2 k − 1)(2 l − 1 ) ˜ t 2 k − 1 ˜ t 2 k − 1 ≡ L − n and L − n τ q = 0 , n = 1 , 2 , 3 , · · · STRING EQU A TIONS OF THE Q- KP H IERARCHY 9 as w e exp ected. By a straightforw ard and t edious calculation, the Vira soro comm utatio n rela- tions [ L − n , L − m ] = ( − n + m ) L − ( n + m ) , m, n = 1 , 2 , 3 , · · · can b e ve rified.  Remark 1. As w e kno w, the q -deformed KP hierarch y reduces to the classical KP hier- arc h y when q → 1 and u 0 = 0. The parameters ( ˜ t 1 , ˜ t 2 , · · · , ˜ t i , · · · ) in eq. (4.6) tend to ( t 1 + x, t 2 , · · · , t i , · · · ) as q → 1. One can furt her identify t 1 + x with x in the classical KP hierarc h y , i.e. t 1 + x → x , therefore the Virasoro generators L − n in eq. (4.6) o f the 2-reduced q -K P hierarc hy tend to ˆ L − n = 1 2 ∞ X i =2 n +1 i 6 =0(mod 2) it i ∂ ∂ t i − 2 n + 1 4 X k + l = n +1 (2 k − 1)(2 l − 1 ) t 2 k − 1 t 2 k − 1 , n = 2 , 3 , · · · (4.13) and ˆ L − 1 = 1 2 ∞ X i =3 i 6 =0(mod 2) it i ∂ ∂ t i − 2 + 1 4 x 2 , (4.14) whic h are iden tical with the r esults of the classical KP hierarc hy giv en by L.A.Dic ke y [29 ] and S.P anda, S.Ro y [26 ]. 5. Conclusions and discussions T o summarize, w e ha ve deriv ed the string equations in eq. (3 .4 ) and the negativ e Virasoro constrain t generators on the τ function of 2 -reduced q -KP hierarc hy in eq. (4.8) in Theorem 2. The results of this pap er sho w ob viously that the Virasoro generators { L − n , n ≥ 1 } of the q -K P hierarc h y are differen t with the { ˆ L − n , n ≥ 1 } of the KP hierarch y , although they satisfy the common Virasoro commutation relations. F urthermore, one can find the following in teresting relation b et w een the q -KP hierarc hy a nd the K P hierarch y L − n = ˆ L − n | t i → ˜ t i = t i + (1 − q ) i i (1 − q i ) x i , and it seems to demonstrate tha t q -deformation is a non-uniform transformation fo r co ordinates t i → ˜ t i , whic h is consisten t with results on τ function [11] and t he q -soliton [14 ] of the q -KP hierarc h y . F or the p- reduced ( p ≥ 3) q -KP hierar c h y , whic h is the q -KP hierarch y satisfying the re- duction condition ( L p ) − = 0, w e can obta in ( M L pn +1 ) − = 0. Using the similar tech nique in q -K dV hierarch y , w e can deduce the Virasoro constrain ts on the τ function of t he p-reduced q -K P hierarc h y for p ≥ 3. Moreo v er, for { L n , n ≥ 0 } w e find a subtle p oin t a t the calculatio n of r es ( V i +1 L − i +2 n ), a nd will try t o study it in the future. Ac knowled gments This w ork is supp orted by th e NSF of Chin a u n der Gran t No. 106 71187, 10825 101. 10 KELEI TIAN † , JINGSONG HE ∗ † ‡ , YUCAI S U † AN D Y I CHENG † Reference s [1] A. Klimyk, K . Schm¨ udgen, Quantum gr oups and their repr esntaions(Springer, B erlin, 19 97). [2] V. Kac, P . Cheung, Q uantum calculus(Spr inger-V erlag, New Y o rk, 200 2 ). [3] D.H. Zhang, Q uantum deformation of KdV hierarchies and their inf initely man y conserv ation laws, J.Phys.A26(1993),2 3 89-2 407. [4] Z.Y. W u, D.H. Zhang, Q.R. Zheng, Qua nt um defor mation of KdV hierarchies and their exact solutions: q -deformed solitons, J. Phys. A27(19 94), 530 7 -5312 . [5] J. Mas, M. 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