BLOCK TYPE SYMMETRY OF BIGRADED TODA HIERARCHY ∗ CHUANZHONG LI§†, JINGSONG HE§ , YUCAI SU‡† 2 §Department of Mathematics, Ningbo University, Ningbo, 315211 Zhejiang, P. R. China 1 †Department of Mathematics, USTC, Hefei, 230026 Anhui, P. R. China 0 ‡Department of Mathematics, Tongji University, Shanghai, 200092, P. R. China 2 n a Abstract. In this paper, we define Orlov-Schulman’soperators ML, MR, and then use them J toconstructtheadditionalsymmetriesofthebigradedTodahierarchy(BTH).Wefurthershow 1 thattheseadditionalsymmetriesformaninterestinginfinitedimensionalLiealgebraknownasa 2 BlocktypeLiealgebra,whosestructuretheoryandrepresentationtheoryhaverecentlyreceived much attention in literature. By acting on two different spaces under the weak W-constraints ] h we find in particular two representations of this Block type Lie algebra. p - h t a Mathematics Subject Classifications (2000). 37K05, 37K10, 37K20, 17B65, 17B67. m Keywords: bigraded Toda hierarchy, additional symmetry, Block type Lie algebra. [ 2 v 4 Contents 8 9 1. Introduction 1 1 1. 2. The two-dimensional Toda hierarchy and its reductions 3 0 3. Bigraded Toda hierarchy 4 2 1 4. Orlov-Schulman’s M , M operators 8 L R : v 5. Additional symmetries of BTH 12 i X 6. Block type actions on functions P and P 17 L R r a 7. Conclusions and discussions 22 References 25 8. Appendix 26 1. Introduction The Toda lattice equation as a completely integrable system was introduced by Toda [1] to describe an infinite system of masses on a line that interact through an exponential force. ∗ Corresponding author: [email protected], [email protected]. 1 2 CHUANZHONGLI,JINGSONGHE, YUCAI SU Inspired by the Sato theory on the Kadomtsev-Petviashvili (KP) hierarchy [2, 3], the two di- mensional Toda hierarchy was constructed by Ueno and Takasaki [4] with the help of difference operators and infinite dimensional Lie algebras. The one dimensional Toda hierarchy (TH) was also studied under the reduction condition L + L−1 = M + M−1 [4] or L = M on two Lax operators L and M. The bigraded Toda hierarchy (BTH) of (N,M)-type(or simply the (N,M)-BTH) is the generalized Toda hierarchy whose infinite Lax matrix has N upper and M lower nonzero diagonals[5]. After continuous interpolation, N and M correspond to the highest and lowest powers of Laurent polynomials which is the Lax operator. The (N,M)- BTH can be naturally considered as a reduction of the two-dimensional Toda hierarchy by imposing an algebraic relation to those two Lax operators(see [4, 5]). One dimensional Toda hierarchy(i.e. (1,1)-BTH) and the BTH have been shown to be related to many mathemat- ical and physical fields such as the inverse scattering method, finite and infinite dimensional algebras, classical and quantum field theories and so on. Recently, in [6, 7], the interpolated Toda lattice hierarchy was generalized to the so-called extended Toda hierarchy (ETH) for considering its application on the topological field theory. In [8], the TH and ETH were further generalized to the extended bigraded Toda hierarchy (EBTH) by considering N+M dependent variables {u ,u ,··· ,u ,u ,u ,··· ,u } in the Lax operator L. This new model has N−1 N−2 1 0 −1, −M been expected [8] that it might be relevant for applications in describing the Gromov-Witten invariants. In fact the dispersionless case of that model has been proposed in [9] because the dispersionless EBTH can be obtained from the dispersionless KP hierarchy. In [10], the Hirota bilinear equations (HBEs) of EBTH have been given conjecturally and proved that it governs the Gromov-Witten theory of orbifold c . In [11], the authors generalize the Sato theory to km the EBTH and give the Hirota bilinear equations in terms of vertex operators whose coefficients take values in the algebra of differential operators. In [12], a geometric structure associated with Frobenius manifold of 2D Toda hierarchy was introduced. Furthermore, motivated by the potential applications of the BTH, which is also defined by omitting the extended logarithmic flows of the EBTH, in the theory of the matrix models, it is necessary and interesting to explore its algebraic structure from the point of view of the additional symmetry. Additional symmetries of KP hierarchy were given by Orlov and Shulman [13] through two novel operators Γ and M, which can be used to form a centerless W algebra. Based on this work, there exist many extensive results (e.g., [14–22]) on the additional symmetries of the KP hierarchy, Toda hierarchy, 2-D Toda hierarchy, BKP hierarchy and CKP hierarchy. Particularly, the representations of the infinite dimensional Virasoro algebra and W algebra havebeenderivedbyusingtheactionsontheLaxoperator,thewavefunctionandtheτ function of additional symmetry flows. These results inspire us to search new infinite dimensional algebras from the additional symmetry flows of the BTH. So the purpose of this paper is to give the additional symmetries of the BTH and then identify its algebraic structure. In [23], the additional symmetries of KP hierarchy were generalized to a W algebra. However, 1+∞ the commutative relations of the W algebra are rather complicated. The algebra under 1+∞ consideration in this paper is very simple and elegant, which is an infinite dimensional Lie algebra B, known as a Block type Lie algebra, introduced by Block [24] around 50 years ago. BLOCK TYPE SYMMETRY OF BIGRADED TODA HIERARCHY 3 ThiskindofLiealgebraisaninterestingobjectinthestructuretheoryandrepresentationtheory of Lie algebras, partly due to its close relation with the Virasoro algebra and the Virasoro-like algebra (e.g., [25–31]). To our best knowledge, this is the first time to bring Block type Lie algebras to the integrable system. In particular, we obtain two representations of the Lie algebra B on two spaces of functions P , P respectively (Theorem 6.1), which (to the best L R of our knowledge) are the first known examples of representations of B with the actions of generating elements of B being explicitly given. Another quite different feature of additional symmetries in the BTH is that M , M − M operator for constructing additional flows L R commutes with Lax operator L, i.e., [L,M] = 0. This is a crucial fact to find Block type Lie algebra here. Note that [L,M] = 1 holds for other known integrable systems such as KP hierarchy. The paper is organized as follows. In Section 2, the two dimensional Toda hierarchy and its reductions are introduced explicitly by which we can define the BTH later. In Section 3, the definition of the BTH and its Sato equation are introduced. In Section 4, we define Orlov- Schulman’s M , M operatorsand prove their linear equations. The additional symmetries and L R related equations of the BTH will be given in Section 5, meanwhile we prove that the additional symmetries have a nice structure of a Block type Lie algebra. InSection 6, we give some specific actions of that Block type additional flows on spaces of functions P , P which further lead to L R representations of this Block type Lie algebra under so-called weak W-constraints. Section 7 is devoted to conclusions and discussions. 2. The two-dimensional Toda hierarchy and its reductions In this section, we will show that the BTH is just a general reduction of the two-dimensional Toda hierarchy whose special reduction leads to original Toda hierarchy. Firstly we will introduce the definition of the two-dimensional Toda hierarchy in interpolated form as following. The two-dimensional Toda hierarchy [4] can be defined by the following two Lax operators, L = Λ+a +a Λ−1 +a Λ−2 +..., (2.1) 0 −1 −2 L¯ = a¯ Λ−1 +a¯ +a¯ Λ1 +a¯ Λ2 +..., (2.2) −1 0 1 2 where Λ represents the shift operator with Λ := eǫ∂x and “ǫ” is called the string coupling constant, i.e. for any function f(x) Λf(x) = f(x+ǫ). The coefficients a and a¯ are the functions of x and {(x ,y ) : n = 1,2,...}. Then the Lax n k n n representation of the two-dimensional Toda hierarchy is given by the set of infinite number of equations for n = 1,2,..., ∂L ∂L = [Ln,L], = [L¯n,L], (2.3) ∂x + ∂y − n n 4 CHUANZHONGLI,JINGSONGHE, YUCAI SU ∂L¯ ∂L¯ = [Ln,L¯], = [L¯n,L¯], (2.4) ∂x + ∂y − n n where Ln represents the part of Ln with non-negative powers in Λ, and L¯n represents the part + − of L¯n with negative powers in Λ. In particular, the Lax equations for n = 1 provide the system for (a ,a¯ ), 0 −1 ∂a¯ (x) −1 = a¯ (x)(a (x)−a (x−ǫ)), ∂x −1 0 0  1 (2.5) ∂a (x)  0 = a¯ (x)−a¯ (x+ǫ).  ∂y −1 −1 1 Withthefunctionu(x)definedbya0(x) = ∂x1u(x)anda¯−1(x) = eu(x)−u(x−ǫ),thetwo-dimensional Toda equation, ∂2u(x) = eu(x)−u(x−ǫ) −eu(x+ǫ)−u(x) (2.6) ∂x ∂y 1 1 is given. The 1-D Toda equation is given by the reduction, L = L¯ = Λ+a +a Λ−1. (2.7) 0 −1 Then the Lax equations for x and y give 1 1 ∂ ∂ + L = [L +L¯ ,L] = [L,L] = 0. (2.8) ∂x ∂y + − (cid:18) 1 1(cid:19) This implies that L does not depend on the variable s := x +y . Then the two-dimensional 1 1 1 Toda equation is reduced to the 1-D Toda equation, and with t := x −y ,u¯(x) = −u(x), we 1 1 1 have the standard 1-D Toda equation as ∂2u¯(x) = eu¯(x−ǫ)−u¯(x) −eu¯(x)−u¯(x+ǫ). (2.9) ∂t2 1 In the next subsection, we generalize the reduction (2.7) to the general Laurent polynomial, i.e. LN = L¯M, N,M ∈ N, (2.10) which defines the (N,M)-BTH(L and L¯ will correspond to fractional powers of Lax operator of the BTH (see (3.14))). 3. Bigraded Toda hierarchy The Lax form of the BTH(i.e. (N,M)-BTH) can be introduced as [8]. For that we need to introduce firstly the Lax operator L = ΛN +u ΛN−1 +···+u Λ−M (3.1) N−1 −M BLOCK TYPE SYMMETRY OF BIGRADED TODA HIERARCHY 5 (where N,M ≥ 1 are two fixed positive integers). The variables u are functions of the real j variable x. The Lax operator L can be written in two different ways by dressing the shift operator L = P ΛNP−1 = P Λ−MP−1. (3.2) L L R R Equation (3.2) is quite important because it gives the reduction condition (2.10) of the BTH from the two-dimensional Toda hierarchy. The two dressing operators have the following form P = 1+w Λ−1 +w Λ−2 +..., (3.3) L 1 2 P = w˜ +w˜ Λ+w˜ Λ2 +..., (3.4) R 0 1 2 and their inverses have form P−1 = 1+Λ−1w′ +Λ−2w′ +..., (3.5) L 1 2 P−1 = w˜′ +Λw˜′ +Λ2w˜′ +.... (3.6) R 0 1 2 The coefficients {w ,w˜,w′,w˜′,i ≥ 0} will be used more later in calculating the representation i i i i of Block algebra. The pair is unique up to multiplying P and P from the right by operators L R in the form 1 + a Λ−1 + a Λ−2 + ... and a˜ + a˜ Λ + a˜ Λ2 + ... respectively with coefficients 1 2 0 1 2 independent of x. From the first identity of (3.2), the relations of u ,−M ≤ i ≤ N − 1 and i w ,j ≥ 1 are as follows (see [11]) j u = w (x)−w (x+Nǫ), (3.7) N−1 1 1 u = w (x)−w (x+Nǫ)−(w (x)−w (x+Nǫ))w (x+(N−1)ǫ), (3.8) N−2 2 2 1 1 1 u = w (x)−w (x+Nǫ) N−3 3 3 −[w (x)−w (x+Nǫ)−(w (x)−w (x+Nǫ))w (x+(N−1)ǫ)]w (x+(N−2)ǫ) 2 2 1 1 1 1 −(w (x)−w (x+Nǫ))w (x+(N−1)ǫ), (3.9) 1 1 2 ········· Moreover, by using the second identity of (3.2), we can also easily get the relations of u and i w˜ formally as follows j w˜ (x) u = 0 , (3.10) −M w˜ (x−Mǫ) 0 w˜ (x)− w˜0(x) w˜ (x−Mǫ) u = 1 w˜0(x−Mǫ) 1 , (3.11) −M+1 w˜ (x−(M −1)ǫ) 0 w˜ (x)− w˜0(x) w˜ (x−Mǫ)− w˜1(x)−w˜0w(˜x0−(xM)ǫ)w˜1(x−Mǫ)w˜ (x−(M −1)ǫ) u = 2 w˜0(x−Mǫ) 2 w˜0(x−(M−1)ǫ) 1 , (3.12) −M+2 w˜ (x−(M −2)ǫ) 0 ··· ··· ··· 6 CHUANZHONGLI,JINGSONGHE, YUCAI SU These relations above will be used in the calculation later. Given any difference operator A = A Λk, the positive and negative projections are given by A = A Λk and k k + k≥0 k A = A Λk. − Pk<0 k P 1 1 To wPrite out explicitly the Lax equations of the BTH, fractional powers LN and LM are defined by LN1 = Λ+ akΛk, LM1 = bkΛk, k≤0 k≥−1 X X with the relations (LN1 )N = (LM1 )M = L. (3.13) Acting on free function λxǫ, these two fraction powers can be seen as two different locally 1 1 expansions around zero and infinity respectively. It was stressed that LN and LM are two different operators even if N = M(N,M ≥ 2) in [8] due to two different dressing operators. They can also be expressed as following LN1 = PLΛPL−1, LM1 = PRΛ−1PR−1. (3.14) Similar to [8], the BTH can be defined as following. Definition 3.1. The bigraded Toda hierarchy consists of a system of flows given in the Lax pair formalism by ∂ L = [A ,L] (3.15) tγ,n γ,n for γ = N,N −1,N −2,...,−M +1 and n ≥ 0. The operators A are defined by γ,n Aγ,n = (Ln+1−γN−1)+ for γ = N,N −1,...,2,1, (3.16a) Aγ,n = −(Ln+1+Mγ )− for γ = 0,−1,...,−M +1, (3.16b) and ∂ is defined as ∂ . tγ,n ∂tγ,n The only difference from [8] is that we cancel the extended flows and add the flow when γ = 1. The flow when γ = 1 is in fact the Toda hierarchy which is also the flow when γ = 0. Particularly for N = 1 = M, this hierarchy coincides with the one dimensional Toda hierar- chy. When N = 1,M = 2, the BTH leads the following primary equations ∂ L = [Λ+u ,L], (3.17) 1,0 0 and ∂ L = −[e(1+Λ−1)−1logu−2Λ−1,L], (3.18) −1,0 which further lead to ∂ u (x) = u (x+ǫ)−u (x), 1,0 0 −1 −1 ∂ u (x) = u (x+ǫ)−u (x)+u (x)(u (x)−u (x−ǫ)), (3.19)  1,0 −1 −2 −2 −1 0 0 ∂1,0u−2(x) = u−2(x)(u0(x)−u0(x−2ǫ)),   BLOCK TYPE SYMMETRY OF BIGRADED TODA HIERARCHY 7 and ∂ u (x) = e(1+Λ−1)−1logu−2(x+ǫ) −e(1+Λ−1)−1logu−2(x), −1,0 0 ∂−1,0u−1(x) = e(1+Λ−1)−1logu−2(x)(u0(x)−u0(x−ǫ)), (3.20) ∂−1,0u−2(x) = u−1(x)e(1+Λ−1)−1logu−2(x−ǫ) −e(1+Λ−1)−1logu−2(x)u−1(x−ǫ). Obviouslyequation(3.20)containsinfinitemultiplicationbecauseofnonlocalterm(1+Λ−1)−1logu (x)  −2 which comes from the fractional power of the Lax operator. Set N = 2 and M = 1, the equations (3.15) are as follows ∂ L = [Λ+(1+Λ)−1u (x),L], (3.21) 2,0 1 ∂ L = [Λ2 +u Λ+u ,L], (3.22) 1,0 1 0 which further lead to the following concrete equations ∂ u (x) = u (x+ǫ)−u (x)+u (x)(1−Λ)(1+Λ)−1u (x), 2,0 1 1 1 1 1 ∂ u (x) = u (x+ǫ)−u (x), (3.23)  2,0 0 −1 −1 ∂2,0u−1(x) = u−1(x)(1−Λ−1)(1+Λ)−1u1(x),   ∂ u (x) = u (x+2ǫ)−u (x), 1,0 1 −1 −1 ∂ u (x) = u (x+2ǫ)−u (x)+u (x)u (x+ǫ)−u u (x−ǫ), (3.24)  1,0 0 −2 −2 1 −1 −1 1 ∂1,0u−1(x) = u−1(x)(u0(x)−u0(x−ǫ)). Notice that the nonlocal term (1 + Λ)−1u (x) also comes from the fractional power of Lax  1 operator in equations above. Therefore appearance of nonlocal term is an important property of the BTH. We can also get more equations when N and M take other integer values but we shall not mention them here because our central consideration in this paper is the Block type additional symmetries of the BTH. For the convenience to derive the Sato equations, the following operators will be defined as in [8, 11]: Ln+1−γ−1 for γ = N ...1, N B := (3.25) γ,n (Ln+1+Mγ for γ = 0···−M +1. Before introducing the Sato equation, the following proposition [8] need to be given firstly. Proposition 3.2. The following two identities hold 1 1 ∂tγ,nLN = [−(Bγ,n)−,LN], (3.26) 1 1 ∂tγ,nLM = [(Bγ,n)+,LM]. (3.27) Proof. See [8, 11]. (cid:3) 8 CHUANZHONGLI,JINGSONGHE, YUCAI SU Using the proposition above, one can obtain the following proposition, lemma and theorem, which are results of [8, 11]. Proposition 3.3. If L satisfies the Lax equation (3.15), then we have the following Zakharov- Shabat equation ∂ (A )−∂ (A )+[A ,A ] = 0, (3.28) tβ,n α,m tα,m β,n α,m β,n for −M +1 ≤ α,β ≤ N , m,n ≥ 0. Using the Zakharov-Shabat equation (3.28) one can obtain the following lemma. Lemma 3.4. ([11]) The following Zakharov-Shabat equations hold ∂ (B ) −∂ (B ) −[(B ) ,(B ) ] = 0, (3.29) β,n α,m − α,m β,n − α,m − β,n − −∂ (B ) +∂ (B ) −[(B ) ,(B ) ] = 0, (3.30) β,n α,m + α,m β,n + α,m + β,n + where, −M +1 ≤ α,β ≤ N, m,n ≥ 0. Using Lemma 3.4 and the Lax equation, one can then obtain the following theorem. Theorem 3.5. ([11]) L is a solution to the BTH if and only if there is a pair of dressing operators P and P , which satisfy the following Sato equations: L R ∂ P = −(B ) P , (3.31) γ,n L γ,n − L ∂ P = (B ) P , (3.32) γ,n R γ,n + R where, −M +1 ≤ γ ≤ N, n ≥ 0. The dressing operators satisfying Sato equations (3.31) and (3.32) will be called wave oper- ators later. After the preparation above, it is time to introduce Orlov-Schulman’s operators which is included in the next section. 4. Orlov-Schulman’s M , M operators L R In order to give the additional symmetries of the BTH, we define the Orlov-Schulman’s M , L M operators by R M = P Γ P−1, M = P Γ P−1, (4.33) L L L L R R R R where N ΓL = x Λ−N + (n+1− α−1)ΛN(n−αN−1)tα,n, (4.34) Nǫ N n≥0 α=1 XX 0 x β ΓR = − ΛM − (n+1+ )Λ−M(n+Mβ )tβ,n. (4.35) Mǫ M n≥0β=−M+1 X X A direct calculation shows that the operators M and M satisfy the following theorem. L R BLOCK TYPE SYMMETRY OF BIGRADED TODA HIERARCHY 9 Proposition 4.1. Operators L,M and M satisfy the following identities L R [L,M ] = 1, [L,M ] = 1, (4.36) L R ∂ M = [A ,M ], ∂ M = [A ,M ], (4.37) tγ,n L γ,n L tγ,n R γ,n R ∂MnLk ∂MnLk L = [A ,MnLk], R = [A ,MnLk], (4.38) ∂t γ,n L ∂t γ,n R γ,n γ,n where, −M +1 ≤ γ ≤ N,n ≥ 0. Proof. Firstly we prove (4.36) by dressing the following two identities using P and P sepa- L R rately N [ΛN,ΓL] = [ΛN, x Λ−N + (n+1− α−1)ΛN(n−αN−1)tα,n] Nǫ N n≥0 α=1 XX = 1, 0 x β [Λ−M,ΓR] = [Λ−M,− ΛM − (n+1+ )Λ−M(n+Mβ )tβ,n] Mǫ M n≥0β=−M+1 X X = 1. For the proof of (4.37), we need to prove ∂ M = [(B ) ,M ], (4.39) tα,n L α,n + L ∂ M = [(B ) ,M ], (4.40) tα,n R α,n + R ∂ M = [−(B ) ,M ], (4.41) t L β,n − L β,n ∂ M = [−(B ) ,M ], (4.42) t R β,n − R β,n where, 1 ≤ α ≤ N,−M +1 ≤ β ≤ 0. Let us consider the following bracket [∂tα,n −ΛN(n+1−αN−1),ΓL] = 0, (4.43) which can be easily got by a direct computation. By dressing both sides of the identity above using the operator P , the left part of (4.43) becomes L [PL∂tα,nPL−1 −Ln+1−αN−1,PLΓLPL−1] = [∂tα,n −(Bα,n)+,ML], which leads to (4.39). For the proof of (4.41), we consider the following bracket [∂ ,Γ ] = 0. (4.44) t L β,n By dressing the identity above in the same way we can get [P ∂ P−1,P Γ P−1] = [∂ +(B ) ,M ], L tβ,n L L L L tβ,n β,n − L 10 CHUANZHONGLI,JINGSONGHE, YUCAI SU which leads to ∂ M = [−(B ) ,M ]. t L β,n − L β,n Similarly, we can prove (4.40) and (4.42) by dressing the identity [∂ ,Γ ] = 0, tα,n R and [∂t +Λ−M(n+1+Mβ ),ΓR] = 0, β,n respectively, through the dressing operator P . Using proved (4.37) and Lax equation (3.15), R (cid:3) we can prove (4.38) easily. The equations in Proposition 4.1 can be realized by linear equations in the following propo- sition. To simplify the theorem, we first introduce two functions w (t,λ) and w (t,λ) which L R have forms w (t,λ) = P (x,Λ)eξL(t,λ) = P (x,Λ)eξL(t,λ), (4.45) L L L w (t,λ) = P (x,Λ)eξR(t,λ) = P (x,Λ)eξR(t,λ), (4.46) R R R where N x ξL(t,λ) = λ(n+1−αN−1)tα,n + logλ, (4.47) Nǫ n≥0 α=1 XX 0 x ξR(t,λ) = − λ−(n+1+Mβ )tβ,n + logλ. (4.48) Mǫ n≥0β=−M+1 X X Functions w (t,λ) and w (t,λ) will be called wave functions. P and P are called symbols of L R L R P and P respectively. L R Proposition 4.2. The wave functions w (t,λ) and w (t,λ) satisfy the following linear equa- L R tions Lw (t,λ) = λw (t,λ), L L M w (t,λ) = ∂ w (t,λ), (4.49)  L L λ L ∂tγ,nwL(t,λ) = Aγ,nwL(t,λ),   Lw (t,λ) = λ−1w (t,λ), R R MRwR(t,λ) = ∂λ−1wR(t,λ), (4.50) ∂tγ,nwR(t,λ) = Aγ,nwR(t,λ), where, −M +1 ≤ γ ≤ N,n ≥0. 