Block algebra in two-component BKP and D type Drinfeld-Sokolov hierarchies
aa r X i v : . [ n li n . S I] O c t BLOCK ALGEBRA IN TWO-COMPONENT BKP AND DTYPE DRINFELD-SOKOLOV HIERARCHIES
CHUANZHONG LI † , JINGSONG HE ‡ Department of Mathematics, Ningbo University, Ningbo, 315211, China † [email protected] ‡ [email protected] Abstract.
We construct generalized additional symmetries of a two-componentBKP hierarchy defined by two pseudo-differential Lax operators. These ad-ditional symmetry flows form a Block type algebra with some modified(oradditional) terms because of a B type reduction condition of this integrablehierarchy. Further we show that the D type Drinfeld-Sokolov hierarchy,which is a reduction of the two-component BKP hierarchy, possess a com-plete Block type additional symmetry algebra. That D type Drinfeld-Sokolovhierarchy has a similar algebraic structure as the bigraded Toda hierarchywhich is a differential-discrete integrable system.
Mathematics Subject Classifications (2000). 37K05, 37K10, 37K20, 17B65,17B67.
Key words : Additional symmetry, Block algebra, Drinfeld-Sokolov hierarchyof type D, two-component BKP hierarchy, bigraded Toda hierarchy.1.
Introduction
One interesting topic in the study of integrable hierarchies is to find sym-metry and its recursion relation,and further to identify its algebraic structures.There are already many results in literatures, for example [1]-[4]. Among thesesymmetries, the additional symmetry is a relatively new type and has beenstudied extensively in recent years, which contains dynamic variables explic-itly and does not commutes with each other. Additional symmetries of theKadomtsev-Petviashvili(KP) hierarchy was introduced by Orlov and Shulman[5] which contain one kind of important symmetry called Virasoro symmetry.These symmetries form a centerless W ∞ algebra is closely related to matrixmodel by means of the Virasoro constraint and string equation[6, 7, 8, 9, 10].Two sub-hierarchies of KP, BKP hierarchy and CKP hierarchy[11, 12], havebeen shown to possess additional symmetry[13]-[14] with consideration of thereductions on the Lax operators.The 2-dimensional Toda Lattice(2dTL) hierarchy is introduced by Ueno andTakasaki in [15] based on the Sato theory. It is natural to construct the addi-tional symmetry of the 2dTL hierarchy because of the similarity between the KPhierarchy and 2dTL hierarchy[10]. For the dispersionless Toda hiearchy[16, 17],additional symmetry is used to give string equations and Riemann-Hilbertproblem. Note that, there exist two different sub-hierarchies of 2dTL hier-archy, 2-dimensional B type Toda Lattice (2dBTL) and 2-dimensional C type ‡ Corresponding author. † , JINGSONG HE ‡ Toda Lattice (2dCTL) [15] which correspond to the infinite-dimensional alge-bras w B ∞ × w B ∞ and w C ∞ × w C ∞ . The additional symmetry of the 2dBTL and2dCTL hierarchies have been given recently in [18]. These results show thatadditional symmetry is one kind of general features of the integrable hierarchies.As a generalization of Virasoro algebra, Block type infinite-dimensional Lie al-gebra and its representation theory have been studied intensively in references[19]-[21]. The Block type Lie algebra ¯ B without cental extension is defined as¯ B = span { L m,l , m, l ∈ Z , l ≥ } , (1.1)with bracket [ L m,l , L n,k ] = ( mk − nl ) L m + n − ,l + k − . (1.2)Note that the Virasoro algebra is one kind of widely used infinite-dimensionalalgebra in mathematical physics, particularly in integrable systems[22]. How-ever, it is curious to note that, in the past 50 years after the introduction of theBlock algebra, there does not exist a result on the application of this algebra inintegrable systems until last year, to the best of our knowledge. In paper[23],we provide a novel Block type algebraic structure of the bigraded Toda hierar-chy(BTH) with the help of the additional symmetry. This is the first time tofind the direct relation between integrable hierarchy and the Block type algebra.Here BTH as a general reduction of 2dTL hierarchy, is introduced in [24, 25]from the background of the topological field and Gromov-Witten invariants.The Hirota bilinear equations and solutions of the BTH are given in [26, 27].Later Block algebra is found again in dispersionless bigraded Toda hierarchy[28], in two-dimensional Toda hierarchy[29]. Very recently Block algebra hasbeen shown to have a close relation with 3-algebra[30].Based on the above results of Block algebra, in order to explore the univer-sality of the Block type algebra in integrable systems, it is necessary to findthis kind of algebraic structure in the KP type differential systems, due to theimportance of the KP systems. In [31], one kind of additional symmetry ofthe KP hierarchy composed one generalized W ∞ algebra with complicatedstructure coefficients but it was not a Block algebra. Taking into considerationthe complexity of the relevant formula of the Block algebra, as well as plentyof different possible extensions of the KP hierarchy, it is a challenging problemto find this algebra in the KP type hierarchies.It is a direct idea to consider the multi-component KP hierarchy(mcKP)[32,33, 34, 35], which has a sole Lax operator with matrix coefficients. However, thealgebra structure of the additional symmetry of the mcKP is very complicatedand belongs to the Virasoro type[36]. Note that integrable systems possessingthe symmetry of the Block type need to have two independent hierarchies offlows defined by two different Lax operators[23, 28, 29]. But the flows of themcKP hierarchy with two pseudo-differential Lax operators are not well defined.Fortunately, for a two-component BKP hierarchy[12, 37], two Lax operatorshas been constructed in [38](see eq.(3.3) and eq.(3.9) of this reference) fromthe view of the Drinfeld-Sokolov Hierarchies of D Type. The Hamiltonianstructure of this two-component BKP hierarchy is given in [39]. Therefore thistwo component BKP hierarchy [38] is a good candidate for us to explore theBlock algebra in integrable hierarchy of KP type. In the following text of thispaper we construct the generalized additional symmetries of the two-componentBKP hierarchy and identify its algebraic structure by using a similar method LOCK ALGEBRA IN TWO-COMPONENT BKP AND D TYPE DRINFELD-SOKOLOV HIERARCHIES3 in [29, 40]. Besides, the D type Drinfeld-Sokolov hierarchy is found to be agood differential model to derive complete Block type infinite dimensional Liealgebra.This paper is arranged as follows. In next section we recall some necessaryfacts of the two-component BKP hierarchy. In Sections 3, we will give thegeneralized additional symmetries for the two-component BKP hierarchy. Byreducing the two-component BKP hierarchy to the D type Drinfeld-Sokolovhierarchy, some concepts and results about this reduced hierarchy will be in-troduced in Section 4. The Block symmetries of Drinfeld-Sokolov hierarchy oftype D will be derived in Section 5.2.
Two component BKP hierarchy
Let us firstly recall some basic facts[38, 41] of the two-component BKP hier-archy which is well defined by two Lax operators. A is assumed as an algebra of smooth functions of a spatial coordinate x andderivation denoted as D = d / d x . This algebra A has following multiplying rule D i · f = X r ≥ (cid:18) ir (cid:19) D r ( f ) D i − r , f ∈ A . For any operator A = P i ∈ Z f i D i ∈ A , its nonnegative projection, negativeprojection, adjoint operator are respectively defined as A + = X i ≥ f i D i , A − = X i< f i D i , A ∗ = X i ∈ Z ( − D ) i · f i . (2.1)Basing on definition in[38], the two Lax operators of the two-component BKPhierarchy have form L = D + X i ≥ u i D − i , ˆ L = D − ˆ u − + X i ≥ ˆ u i D i , (2.2)such that L ∗ = − DLD − ˆ L ∗ = − D ˆ LD − , r ∈ Z + . (2.3)We call eq.(2.3) the B type condition of two-component BKP hierarchy.The two-component BKP hierarchy is defined by the following Lax equations: ∂ ¯ L∂t k = [( L k ) + , ¯ L ] , ∂ ¯ L∂ ˆ t k = [ − ( ˆ L k ) − , ¯ L ] (2.4)with ¯ L = L or ˆ L, k ∈ Z odd+ .Note that ∂/∂t flow is equivalent to ∂/∂x flow, therefore it is reasonable toassume t = x in the following several sections.One can write the operators L and ˆ L in a dressing form as L = Φ D Φ − , ˆ L = ˆΦ D − ˆΦ − , (2.5)where Φ = 1 + X i ≥ a i D − i , ˆΦ = 1 + X i ≥ b i D i (2.6)satisfy Φ ∗ = D Φ − D − , ˆΦ ∗ = D ˆΦ − D − . (2.7) CHUANZHONG LI † , JINGSONG HE ‡ Given L and ˆ L , the dressing operators Φ and ˆΦ are determined uniquely upto a multiplication to the right by operators with constant coefficients. Thetwo-component BKP hierarchy (2.4) can also be redefined as ∂ Φ ∂t k = − ( L k ) − Φ , ∂ ˆΦ ∂t k = (cid:0) ( L k ) + − δ k ˆ L − (cid:1) ˆΦ , (2.8) ∂ Φ ∂ ˆ t k = − ( ˆ L k ) − Φ , ∂ ˆΦ ∂ ˆ t k = ( ˆ L k ) + ˆΦ , (2.9)with k ∈ Z odd+ .Denote t = ( t , t , t , . . . ), ˆ t = (ˆ t , ˆ t , ˆ t , . . . ) and introduce two wave functions w ( z ) = w ( t, ˆ t ; z ) = Φ e ξ ( t ; z ) , ˆ w ( z ) = ˆ w ( t, ˆ t ; z ) = ˆΦ e xz + ξ (ˆ t ; − z − ) , (2.10)where the function ξ is defined as ξ ( t ; z ) = P k ∈ Z odd+ t k z k . It is easy to see D i e xz = z i e xz , i ∈ Z and L w ( z ) = zw ( z ) , ˆ L ˆ w ( z ) = z − ˆ w ( z ) . The two-component BKP hierarchy was proved to have infinitely many bi-Hamiltonian structures and Hamiltonian densities which are the residues of L k and ˆ L k with tau-symmetric condition[39]. The tau function of the two-component BKP hierarchy can be defined in form of the wave functions as w ( t, ˆ t ; z ) = τ ( t − z − ] , ˆ t ) τ ( t, ˆ t ) e ξ ( t ; z ) , ˆ w ( t, ˆ t ; z ) = τ ( t, ˆ t + 2[ z ]) τ ( t, ˆ t ) e ξ (ˆ t ; − z − ) (2.11)where [ z ] = ( z, z / , z / , . . . ).With above preparation, it is time to construct generalized additional sym-metries for the two-component BKP hierarchy in the next section.3. Generalized additional symmetries of the two-componentBKP hierarchy
In this section, we are to construct generalized additional symmetries for thetwo-component BKP hierarchy by using the Orlov–Schulman operators whosecoefficients depend explicitly on the time variables of the hierarchy.With the same dressing operators given in eq.(2.6), Orlov–Schulman opera-tors M, ˆ M are constructed in following dressing structure [5, 41] M = ΦΓΦ − , ˆ M = ˆΦˆΓ ˆΦ − , where Γ = X k ∈ Z odd+ kt k D k − , ˆΓ = x + X k ∈ Z odd+ k ˆ t k D − k − . Then it is easy to get the following lemma.
Lemma 3.1.
The operators M and ˆ M satisfy [ L, M ] = 1 , [ ˆ L − , ˆ M ] = 1; M w ( z ) = ∂ z w ( z ) , ˆ M ˆ w ( z ) = ∂ z ˆ w ( z );(3.1) LOCK ALGEBRA IN TWO-COMPONENT BKP AND D TYPE DRINFELD-SOKOLOV HIERARCHIES5 and for ¯ M = M or ˆ M , ∂ ¯ M∂t k = [( L k ) + , ¯ M ] , ∂ ¯ M∂ ˆ t k = [ − ( ˆ L k ) − , ¯ M ] , k ∈ Z odd+ . (3.2)For an operator A = A ( L, ˆ L − , M, ˆ M ) which can be written as an antisym-metric form as A = B − D − B ∗ D , define a flow Y A acting on Φ and ˆΦ as Y A Φ = − A − Φ , Y A ˆΦ = A + ˆΦ , (3.3)therefore Y A L = [ − A − , L ] , Y A ˆ L = [ A + , ˆ L ] , and Y A M = [ − A − , M ] , Y A ˆ M = [ A + , ˆ M ] . Following calculation can be easily got[ Y A , Y B ]Φ = − ( Y A B ) − Φ + ( Y B A ) − Φ + [ B − , A − ]Φ , [ Y A , Y B ] ˆΦ = ( Y A B ) + ˆΦ − ( Y B A ) + ˆΦ + [ B + , A + ] ˆΦ . Above two identities can be written as an universal form[ Y A , Y B ]Φ( ˆΦ) = Y { B,A } Φ( ˆΦ) , (3.4)where { A, B } = − Y A B + Y B A − [ A − , B − ] + [ A + , B + ] . (3.5)Also one can derive following proposition. Proposition 3.2.
For any polynomial A = A ( L, ˆ L − , M, ˆ M ) , one has ∂A∂t k = [( L k ) + , A ] , ∂A∂ ˆ t k = [ − ( ˆ L k ) − , A ] , k ∈ Z odd+ . (3.6) Proof
Proof is easy to finish by considering eqs.(2.4) and eqs.(3.2). (cid:3)
Using eq.(3.3) and Proposition 3.2, it can be proved that the flow eqs.(3.3)can commute with original flow of the two-component BKP hierarchy, i.e.[ Y A , ∂∂t k ] = 0 , [ Y A , ∂∂ ˆ t k ] = 0 . (3.7)That means they are symmetries of the two-component BKP hierarchy. Thiskind of symmetries contain original additional w B ∞ × w B ∞ symmetry of the two-component BKP hierarchy mentioned in [41]. The definition of operator A here is more general than operators used to construct additional symmetryof two component BKP hierarchy in [41] because the multiplication mixed set { L, M } and set { ˆ L, ˆ M } together. Therefore we call it the generalized additionalsymmetry of the two-component BKP hierarchy. Here we will not give a detailedproof of this symmetry but later we will prove some special symmetry of thiskind of generalized additional symmetries.The new bracket structure { , } can be expressed by the standard bracketstructure [ , ] which is showed in the following lemma. CHUANZHONG LI † , JINGSONG HE ‡ Lemma 3.3.
Following relations between two bracket structure hold { f ˆ f , g ˆ g } = [ f, g ] ˆ f ˆ g − f g [ ˆ f , ˆ g ] , (3.8) { ˆ f f, ˆ gg } = ˆ f ˆ g [ f, g ] − [ ˆ f , ˆ g ] f g, (3.9) { f ˆ f , ˆ gg } = [ f, ˆ g ][ g, ˆ f ] − f [ ˆ f , ˆ g ] g + ˆ g [ f, g ] ˆ f , (3.10) where f, g are polynomials of L, M and ˆ f , ˆ g are polynomials of ˆ L, ˆ M .
Proof
The first two identities can be easily derived by direct calculation basingon definition, therefore we only give the proof of the identity (3.10) as following { f ˆ f , ˆ gg } = − Y f ˆ f (ˆ gg ) + Y ˆ gg ( f ˆ f ) + [( f ˆ f ) + , (ˆ gg ) + ] − [( f ˆ f ) − , (ˆ gg ) − ]= − [( f ˆ f ) + , ˆ g ] g + ˆ g [( f ˆ f ) − , g ] − [(ˆ gg ) − , f ] ˆ f + f [(ˆ gg ) + , ˆ f ]+ [( f ˆ f ) + , (ˆ gg ) + ] − [( f ˆ f ) − , (ˆ gg ) − ]= f ˆ gg ˆ f + ˆ gf ˆ f g − f ˆ f ˆ gg − ˆ ggf ˆ f = ( f ˆ g − ˆ gf )( g ˆ f − ˆ f g ) − f ˆ f ˆ gg + f ˆ g ˆ f g + ˆ gf g ˆ f − ˆ ggf ˆ f = [ f, ˆ g ][ g, ˆ f ] − f [ ˆ f , ˆ g ] g + ˆ g [ f, g ] ˆ f . (cid:3) From eq.(3.4), it is easy to see following lemma holds.
Lemma 3.4.
There is an antihomorphism between two sets, i.e. C [ L, ˆ L, M, ˆ M ] and G = { Y A | A = A ( L, ˆ L − , M, ˆ M ) } C [ L, ˆ L, M, ˆ M ] , { } 7→ , G , [ ] ,A , Y A , which satisfy following antihomorphism relation [ Y A , Y B ]Φ( ˆΦ) = Y { B,A } Φ( ˆΦ) . Because of the anti-order of spectral representation of multiplications of Laxoperators and Orlov-Schulman operators, following lemmas can be easily de-rived.
Lemma 3.5.
For a , a , b , b ∈ Z + , there is an anti homorphism ω ∞ ⊗ ω ∞ , [ ] C [ L, ˆ L − , M, ˆ M ] , { } ,z a ∂ b z z a ∂ b z M b L a ˆ L − a ˆ M b , [ z a ∂ b z z a ∂ b z , z c ∂ d z z c ∂ d z ]
7→ { M d L c ˆ L − d ˆ M d , M b L a ˆ L − a ˆ M b } . Lemma 3.6.
For a , a , b , b ∈ Z + , there is anisomorphism ψ : ω ∞ ⊗ ω ∞ , [ ]
7→ G , [ ] ,z a ∂ b z z a ∂ b z Y M b L a ˆ L − a ˆ M b , [ z a ∂ b z z a ∂ b z , z c ∂ d z z c ∂ d z ] Y { M d L c ˆ L − d ˆ M d ,M b L a ˆ L − a ˆ M b } , i.e. [ ψ ( z a ∂ b z z a ∂ b z ) , ψ ( z c ∂ c z z d ∂ d z )] = ψ ([ z a ∂ b z z a ∂ b z , z c ∂ d z z d ∂ d z ]) . (3.11) LOCK ALGEBRA IN TWO-COMPONENT BKP AND D TYPE DRINFELD-SOKOLOV HIERARCHIES7
From now on, we will introduce one special kind of case of A = A ( L, ˆ L − , M, ˆ M ),i.e. the following two operators B ml and ˆ B ml . Given any pair of integers ( m, l )with m, l ≥
0, define B ml = M L m +1 ˆ L − l + ( − l + m ˆ L − l L m M L, (3.12)ˆ B ml = L m ˆ L − l +1 ˆ M + ( − l + m ˆ L ˆ M ˆ L − l L m . (3.13)The definitions of B ml and ˆ B ml are also different from definitions in [41]. As acorollary of Proposition 3.2, following proposition can be got. Proposition 3.7.
For any ¯ B ml = B ml , ˆ B ml , one has ∂ ¯ B ml ∂t k = [( L k ) + , ¯ B ml ] , ∂ ¯ B ml ∂ ˆ t k = [ − ( ˆ L k ) − , ¯ B ml ] , k ∈ Z odd+ . (3.14)To prove that B ml and ˆ B ml satisfy B type condition, we need following lemma. Lemma 3.8.
Operators M and ˆ M satisfy following conjugate identities, M ∗ = DL − M LD − , ˆ M ∗ = D ˆ L ˆ M ˆ L − D − . (3.15) Proof
UsingΦ ∗ = D Φ − D − , ˆΦ ∗ = D ˆΦ − D − , following calculations M ∗ = Φ ∗− ΓΦ ∗ = D Φ D − Γ D Φ − D − = D Φ D − Φ − M Φ D Φ − D − , ˆ M ∗ = ˆΦ ∗− ˆΓ ˆΦ ∗ = D ˆΦ D − ˆΓ D ˆΦ − D − = D ˆΦ D − ˆΦ − ˆ M ˆΦ D ˆΦ − D − , will lead to this lemma. (cid:3) It is easy to check following proposition holds basing on the Lemma 3.8 above.
Proposition 3.9. B ml and ˆ B ml satisfy B type condition, namely B ∗ ml = − DB ml D − , ˆ B ∗ ml = − D ˆ B ml D − . (3.16) Proof
Using Proposition 3.8, following calculation will lead to first identity ofthis proposition B ∗ ml = ( M L m +1 ˆ L − l − ( − l + m +1 ˆ L − l L m M L ) ∗ = ˆ L − l ∗ L m +1 ∗ M ∗ − ( − l + m +1 L ∗ M ∗ L m ∗ ˆ L − l ∗ = ( − l + m +1 D ˆ L − l L m M LD − − DM L m +1 ˆ L − l D − = − D ( M L m +1 ˆ L − l − ( − l + m +1 ˆ L − l L m M L ) D − . The second identity can be proved in similar way. (cid:3)
Because of Proposition 3.9, the following equations are well defined ∂ Φ ∂b ml = Y B ml Φ = − ( B ml ) − Φ , ∂ ˆΦ ∂b ml = Y B ml ˆΦ = ( B ml ) + ˆΦ , (3.17) ∂ Φ ∂ ˆ b ml = Y ˆ B ml Φ = − ( ˆ B ml ) − Φ , ∂ ˆΦ ∂ ˆ b ml = Y ˆ B ml ˆΦ = ( ˆ B ml ) + ˆΦ . (3.18) CHUANZHONG LI † , JINGSONG HE ‡ These equations are equivalent to following Lax equations ∂L∂b ml = [ − ( B ml ) − , L ] , ∂ ˆ L∂b ml = [( B ml ) + , ˆ L ] , (3.19) ∂L∂ ˆ b ml = [ − ( ˆ B ml ) − , L ] , ∂ ˆ L∂ ˆ b ml = [( ˆ B ml ) + , ˆ L ] . (3.20)These flows are in fact some special cases of generalized additional symme-tries eqs.(3.3). To show some techniques in the proof of generalized symmetryeq.(3.7), we will give a short proof of following proposition. Proposition 3.10.
The flows (3.19) and (3.20) commute with the flows of thetwo-component BKP hierarchy. Namely, for any ¯ b ml = b ml , ˆ b ml and ¯ t k = t k , ˆ t k one has (cid:20) ∂∂ ¯ b ml , ∂∂ ¯ t k (cid:21) = 0 , m, l ∈ Z + , k ∈ Z odd+ , (3.21) which holds in the sense of acting on Φ or ˆΦ .Proof The proposition can be checked case by case with the help of eq.(3.14)and eqs.(3.19)-(3.20). For example, (cid:20) ∂∂b ml , ∂∂ ˆ t k (cid:21) Φ=[( ˆ L k ) − , ( B ml ) − ]Φ − [( B ml ) + , ˆ L k ] − Φ − [( ˆ L k ) − , B ml ] − Φ = 0 , (cid:20) ∂∂ ˆ b ml , ∂∂t k (cid:21) ˆΦ=[( L k ) + − δ k ˆ L − , ( ˆ B ml ) + ] ˆΦ + (cid:16) [ − ( ˆ B ml ) − , L k ] + − δ k [( ˆ B ml ) + , ˆ L − ] (cid:17) ˆΦ − [( L k ) + , ˆ B ml ] + ˆΦ = 0 . The other cases can be proved in similar ways. This is the end of this proposi-tion. (cid:3)
This proposition implies that the additional flows (3.19)-(3.20) are symme-tries of the two-component BKP hierarchy. To see the further structure of theadditional symmetry, we need following proposition.
Proposition 3.11.
The operators ¯ B m,n = B m,n , ˆ B m,n , m, n ∈ Z + of the two-component BKP hierarchy satisfy following identity { B m ,m , B n ,n } = ( m − n ) B m + n ,m + n + Q m ,m ,n ,n , { ˆ B m ,m , ˆ B n ,n } = ( m − n ) ˆ B m + n ,m + n + ˆ Q m ,m ,n ,n , { B m ,m , ˆ B n ,n } = − n ˆ B m + n ,m + n + m B m + n ,m + n + ¯ Q m ,m ,n ,n , where Q m ,m ,n ,n = ( − m + m +1 { M L n +1 ˆ L − n , ˆ L − m L m M L } +( − n + n { M L m +1 ˆ L − m , ˆ L − n L n M L } , LOCK ALGEBRA IN TWO-COMPONENT BKP AND D TYPE DRINFELD-SOKOLOV HIERARCHIES9 ˆ Q m ,m ,n ,n = ( − m + m +1 { L n ˆ L − n +1 ˆ M , ˆ L ˆ M ˆ L − m L m } +( − n + n { L m ˆ L − m +1 ˆ M , ˆ L ˆ M ˆ L − n L n } , ¯ Q m ,m ,n ,n = ( − m + m +1 { L n ˆ L − n +1 ˆ M , ˆ L − m L m M L } +( − n + n { M L m +1 ˆ L − m , ˆ L ˆ M ˆ L − n L n } . Take ˜ B m ,m = ( m + 1) B m ,m − m ˆ B m ,m , then { ˜ B m ,m , ˜ B n ,n } = (( n + 1) m − ( m + 1) n ) ˜ B m + n ,m + n + S m ,m ,n ,n , where S m ,m ,n ,n = ( m + 1)( n + 1) Q m ,m ,n ,n − n ( m + 1) ¯ Q m ,m ,n ,n + m ( n + 1) ¯ Q n ,n ,m ,m + m n ˆ Q m ,m ,n ,n . Then the following theorem is clear.
Theorem 3.12.
In the sense of acting on Φ or ˆΦ , the additional flows (3.19) and (3.20) satisfy following relations [ ∂ b m ,m , ∂ b n ,n ] = ( m − n ) ∂ b m n ,m n + Y Q m ,m ,n ,n , (3.22)[ ∂ ˆ b m ,m , ∂ ˆ b n ,n ] = ( m − n ) ∂ ˆ b m n ,m n + Y ˆ Q m ,m ,n ,n , (3.23)[ ∂ b m ,m , ∂ ˆ b n ,n ] = m ∂ b m n ,m n − n ∂ ˆ b m n ,m n + Y ¯ Q m ,m ,n ,n . (3.24) These above relations further lead to following modified Block type algebraicrelation [ ∂ v m ,m , ∂ v n ,n ] = (( n + 1) m − ( m + 1) n ) ∂ v m n ,m n + Y S m ,m ,n ,n , (3.25) where ∂ v m ,m = ( m + 1) ∂ b m ,m − m ∂ ˆ b m ,m . Without B type condition eq.(2.3), the operators ( Q, ˆ Q, S, Y Q , Y ˆ Q , Y S ) willvanish. This will lead to nice Block symmetric structure. Proposition 3.10 andTheorem 3.12 show that the flows (3.19) and (3.20) give one modified Blocktype additional symmetries for the two-component BKP hierarchy. The obsta-cle to derive the perfect Block type symmetry is due to the constrained B typecondition eq.(2.3) of the two-component BKP hierarchy. That means if we onlyconsider the the two-component KP hierarchy, i.e. the two-component BKPhierarchy without the constrained B type condition, the additional symmetrywill compose nice structure of Block type infinite dimensional Lie algebra. Butunfortunately the Lax representation with two different pseudo-differential op-erators of the two-component KP hierarchy is not well-defined.If we choose l = 0 in the operator B m,l , m = 0 in the operator ˆ B m,l andincrease one index on each operator of M, ˆ M , then this symmetry will be the w B ∞ × w B ∞ algebra mentioned in [41]. † , JINGSONG HE ‡ Although we only get modified Block type additional symmetries for the two-component BKP hierarchy, further calculation in the next section supports:If we do a (2n,2)-reduction from the two-component BKP hierarchy, perfectBlock type additional symmetry will be exactly kept. This reduced hierarchy isnothing but the D type Drinfeld–Sokolov hierarchies [42] which will be discussedin the next section.4.
D type Drinfeld–Sokolov hierarchy
Assume a new Lax operator L which has following relation with two Laxoperators of the two-component BKP hierarchy introduced in last section L = L n = ˆ L , n ≥ . (4.1)Then the Lax operators of two-component BKP hierarchy will be reduced tothe following Lax operator of D type Drinfeld–Sokolov hierarchy[38, 41] L = D n + 12 n X i =1 D − (cid:0) v i D i − + D i − v i (cid:1) + D − ρD − ρ. (4.2)The difference of the Lax operator L from the one of the D type Drinfeld–Sokolov hierarchy in [38, 41] is we did a shift on n , i.e. we change n − , n ≥ n, n ≥ Remark:
It seems that one can not compute the square of the operator ˆ L ,because it contains infinite terms with positive powers of D and is not a pseudo-differential operator in common sense. In eqs.(6.1)-(6.2) in [41], Chaozhong Wugive the above reduction directly without a proof because in paper [38] theyhave spent a lot of space to carry out the proof. Here we only describe somekey points of the proof in [38] which is in a inverse direction. The Lemma 3.1and Lemma 3.3 in [38] tell us, for a given operator L in eq.(4.2), there existstwo fractional operators L n and L in same forms as L and ˆ L (eq.(2.2)) indifferent operator rings. In [38], they prove that one can choose two fractionaloperators to be exactly the Lax operators L and ˆ L in the two-component BKPhierarchy in last section because they satisfy all the necessary characters suchas antisymmetric property. The difficulty is in the proof of Lemma 3.3 in [38]with the help of Lemma 2.5 in [38] which promise the reasonability to define thesquare root of the pseudo-differential operator L . Therefore to save the space,we will not give the repeated proof as [38] on the consistency between the Dtype Drinfeld–Sokolov hierarchy and the two-component BKP hierarchy underthe reduction condition eq.(4.1).One can easily find the Lax operator L of D type Drinfeld–Sokolov hierarchywill not satisfy the reduction condition as Lax operator of the two-componentBKP hierarchy but satisfy following B type condition L ∗ = D L D − . (4.3)This Lax operator L of D type Drinfeld–Sokolov hierarchy has following dressingstructure[41] L = Φ D n Φ − = ˆΦ D − ˆΦ − . (4.4)Here Φ = 1 + X i ≥ a i D − i , ˆΦ = 1 + X i ≥ b i D i (4.5) LOCK ALGEBRA IN TWO-COMPONENT BKP AND D TYPE DRINFELD-SOKOLOV HIERARCHIES11 are pseudo-differential operators that also satisfy following B type conditionΦ ∗ = D Φ − D − , ˆΦ ∗ = D ˆΦ − D − . (4.6)The dressing structure inspire us to define two fractional operators as L n = D + X i ≥ u i D − i , L = D − ˆ u − + X i ≥ ˆ u i D i . (4.7)Two fractional operators L n and L can be rewritten in a dressing form as L n = Φ D Φ − , L = ˆΦ D − ˆΦ − . (4.8)The D type Drinfeld–Sokolov hierarchy being considered in this paper isdefined by the following Lax equations: ∂ L ∂t k = [( L k n ) + , L ] , ∂ L ∂ ˆ t k = [ − ( L k ) − , L ] , k ∈ Z odd+ . (4.9)Among these hierarchies, the Drinfeld–Sokolov hierarchy of type D n is asso-ciated to the affine algebra D (1) n and the zeroth vertex of its Dynkin diagram[42, 38]. Similarly as the two-component BKP hierarchy, the equivalence be-tween ∂/∂t and ∂/∂x leads to assumption as t = x .The dressing operators Φ and ˆΦ are same as the ones of two-component BKPhierarchy. Given L , the dressing operators Φ and ˆΦ are uniquely determinedup to a multiplication to the right by operators of the form (4.5) and (4.6)with constant coefficients. The D type Drinfeld-Sokolov hierarchies can also beredefined as ∂ Φ ∂t k = − ( L k n ) − Φ , ∂ ˆΦ ∂t k = (cid:0) ( L k n ) + − δ k L − (cid:1) ˆΦ , (4.10) ∂ Φ ∂ ˆ t k = − ( L k ) − Φ , ∂ ˆΦ ∂ ˆ t k = ( L k ) + ˆΦ (4.11)with k ∈ Z odd+ .Introduce two wave functions w ( z n ) = w ( t, ˆ t ; z n ) = Φ e ξ ( t ; z n ) , (4.12)ˆ w ( z ) = ˆ w ( t, ˆ t ; z ) = ˆΦ e xz + ξ (ˆ t ; − z − ) . (4.13)It is easy to see L w ( z n ) = zw ( z n ) , L ˆ w ( z ) = z − ˆ w ( z ) . After above preparation, we will show that this D type Drinfeld-Sokolovhierarchies have nice Block symmetry as its appearance in BTH [23].5.
Block symmetries of D type Drinfeld-Sokolov hierarchies
In this section, we will put constrained condition eq.(4.1) into constructionof the flows of additional symmetry which form the well-known Block algebra.With the dressing operators given in eq.(4.8), we introduce Orlov-Schulmanoperators as following M = ΦΓ L Φ − , ˆ M = ˆΦˆΓ R ˆΦ − , † , JINGSONG HE ‡ where Γ L = X k ∈ Z odd+ k n t k D k − n , ˆΓ R = − x D − X k ∈ Z odd+ k ˆ t k D − k . It is easy to see the following lemma holds.
Lemma 5.1.
The operators M and ˆ M satisfy [ L , M ] = 1 , [ L , ˆ M ] = 1; (5.1) and M w ( z n ) = ∂ z w ( z n ) , ˆ M ˆ w ( z ) = ∂ z − ˆ w ( z ); (5.2) ∂ ¯ M ∂t k = [( L k n ) + , ¯ M ] , ∂ ¯ M ∂ ˆ t k = [ − ( L k ) − , ¯ M ] , (5.3) where ¯ M = M or ˆ M , k ∈ Z odd+ . To make the operators used in additional symmetry satisfying B type con-dition, we need to prove the following B type property of M − ˆ M which isincluded in following lemma. Lemma 5.2.
The difference of two Orlov-Schulman operators M and ˆ M forD type Drinfeld-Sokolov hierarchy has following D type property: L ∗ ( M − ˆ M ) ∗ = − D L ( M − ˆ M ) D − . (5.4) Proof
It is easy to find the two Orlov-Schulman operators M and ˆ M of the Dtype Drinfeld-Sokolov hierarchy can be expressed by Orlov-Schulman operators M, ˆ M and Lax operators L, ˆ L of two-component BKP hierarchy as M = M L − n n , ˆ M = − ˆ M ˆ L − . (5.5)Using Lemma 3.8, putting eq.(5.5) into ( M − ˆ M ) ∗ can lead to( M − ˆ M ) ∗ = − DL − n M LD − n − D ˆ L − ˆ M ˆ L − D − − DL − n M LD − n − D ˆ L − ˆ M ˆ L − D − − D ( M L − n − nL − n ) D − n − D ( ˆ M ˆ L − + 2 ˆ L − ) D − , (5.8)which can further lead to L ∗ ( M − ˆ M ) ∗ = − D ( LM − L ˆ M ) D − . (5.9)In above calculation, the commutativity between L and M − ˆ M is alreadyused. Till now, the proof is finished. (cid:3) For D-type Drinfeld-Sokolov hierarchy, m is supposed to be odd number toavoid being trivial and simplify it to B m,l = ( M − ˆ M ) m L l , m ∈ Z odd+ . (5.10)One can easily check that B ∗ m,l = − D B m,l D − , m ∈ Z odd+ . (5.11) LOCK ALGEBRA IN TWO-COMPONENT BKP AND D TYPE DRINFELD-SOKOLOV HIERARCHIES13
That means it is reasonable to define additional flow of the D type Drinfeld–Sokolov hierarchy ∂ L ∂c m,l = [ − ( B m,l ) − , L ] , m ∈ Z odd+ , l ∈ Z + . (5.12) Proposition 5.3.
For the Drinfeld–Sokolov hierarchy of type D , the flows (5.12) can commute with original flow of the Drinfeld–Sokolov hierarchy of type D , namely, (cid:20) ∂∂c m,l , ∂∂t k (cid:21) = 0 , (cid:20) ∂∂c m,l , ∂∂ ˆ t k (cid:21) = 0 , l ∈ Z + , m, k ∈ Z odd+ , which hold in the sense of acting on Φ , ˆΦ or L . Proof
According to the definition,[ ∂ c m,l , ∂ t k ]Φ = ∂ c m,l ( ∂ t k Φ) − ∂ t k ( ∂ c m,l Φ) , and using the actions of the additional flows and the flows of D type Drinfeld-Sokolov hierarchy on Φ, we have[ ∂ c m,l , ∂ t k ]Φ = − ∂ c m,l (cid:16) ( L k n ) − Φ (cid:17) + ∂ t k (cid:16) (( M − ˆ M ) m L l ) − Φ (cid:17) = − ( ∂ c m,l L k n ) − Φ − ( L k n ) − ( ∂ c m,l Φ)+[ ∂ t k (( M − ˆ M ) m L l )] − Φ + ((
M − ˆ M ) m L l ) − ( ∂ t k Φ) . Using eq.(4.9) and eq.(5.3), it equals[ ∂ c m,l , ∂ t k ]Φ = [ (cid:16) ( M − ˆ M ) m L l (cid:17) − , L k n ] − Φ + ( L k n ) − (cid:16) ( M − ˆ M ) m L l (cid:17) − Φ+[( L k n ) + , ( M − ˆ M ) m L l ] − Φ − (( M − ˆ M ) m L l ) − ( L k n ) − Φ= [((
M − ˆ M ) m L l ) − , L k n ] − Φ − [( M − ˆ M ) m L l , ( L k n ) + ] − Φ+[( L k n ) − , (( M − ˆ M ) m L l ) − ]Φ= 0 . The other cases of this proposition can be proved in similar ways. (cid:3)
Above proposition indicate that eq.(5.12) is symmetry of D type Drinfeld-Sokolov hierarchy. Further we can get following identities hold ∂ M ∂c m,l = [ − ( B m,l ) − , M ] , ∂ ˆ M ∂c m,l = [( B m,l ) + , ˆ M ] , m ∈ Z odd+ , l ∈ Z + , (5.13) ∂w ( z n ) ∂c m,l = − ( B m,l ) − w ( z n ) , ∂ ˆ w ( z ) ∂c m,l = ( B m,l ) + ˆ w ( z ) , j ≥ − . (5.14)Using same technique used in [23], following theorem can be derived. Theorem 5.4.
The flows in eq. (5.12) about additional symmetries of D typeDrinfeld-Sokolov hierarchy compose following Block type Lie algebra [ ∂ c m,l , ∂ c s,k ] = ( km − sl ) ∂ c m + s − ,k + l − , m, s ∈ Z odd+ , k, l ∈ Z + , † , JINGSONG HE ‡ which holds in the sense of acting on Φ , ˆΦ or L . Proof
By using eq.(5.12) and eq.(5.13), we get[ ∂ c m,l , ∂ c s,k ]Φ = ∂ c m,l ( ∂ c s,k Φ) − ∂ c s,k ( ∂ c m,l Φ)= − ∂ c m,l (cid:16) (( M − ˆ M ) s L k ) − Φ (cid:17) + ∂ c s,k (cid:16) (( M − ˆ M ) m L l ) − Φ (cid:17) = − ( ∂ c m,l ( M − ˆ M ) s L k ) − Φ − (( M − ˆ M ) s L k ) − ( ∂ c m,l Φ)+( ∂ c s,k ( M − ˆ M ) m L l ) − Φ + ((
M − ˆ M ) m L l ) − ( ∂ c s,k Φ) , which further leads to[ ∂ c m,l , ∂ c s,k ]Φ= − h s − X p =0 ( M − ˆ M ) p ( ∂ c m,l ( M − ˆ M ))( M − ˆ M ) s − p − L k + ( M − ˆ M ) s ( ∂ c m,l L k ) i − Φ − (( M − ˆ M ) s L k ) − ( ∂ c m,l Φ)+ h m − X p =0 ( M − ˆ M ) p ( ∂ c s,k ( M − ˆ M ))( M − ˆ M ) m − p − L l + ( M − ˆ M ) m ( ∂ c s,k L l ) i − Φ+((
M − ˆ M ) m L l ) − ( ∂ c s,k Φ)= [( sl − km )( M − ˆ M ) m + s − L k + l − ] − Φ= ( km − sl ) ∂ c m + s − ,k + l − Φ . In the process of deriving the above nice algebraic structure, we omitted a lotof tedious calculation among operators. Similarly the same results on ˆΦ and L can be got. (cid:3) Our early papers and above results show the Block type algebras are ap-peared not only in Toda type difference systems but also in differential systemssuch as two-BKP hierarchy, D type Drinfeld-Sokolov hierarchy, which representsone kind of hidden symmetry algebraic structures of them. These results alsoshow that Block infinite dimensional Lie algebra has a certain of universalityin integrable hierarchies.
Acknowledgments.
We are grateful to Prof. Dafeng Zuo, Jipeng Cheng,Zhiwei Wu and Kelei Tian for valuable discussions. We also thank the refereefor his/her valuable suggestions on the proof of Theorem 5.4. Chuanzhong Li issupported by the National Natural Science Foundation of China under GrantNo. 11201251, the Natural Science Foundation of Zhejiang Province underGrant No. LY12A01007, the Natural Science Foundation of Ningbo under GrantNo. 2013A610105. Jingsong He is supported by the National Natural ScienceFoundation of China under Grant No. 11271210, K.C.Wong Magna Fund inNingbo University.
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