The Additional Symmetries for the BTL and CTL Hierarchies
The Toda lattice (TL) hierarchy was first introduced by K.Ueno and K.Takasaki in \cite{uenotaksasai} to generalize the Toda lattice equations\cite{toda}. Along the work of E. Date, M. Jimbo, M. Kashiwara and T. Miwa \cite{DJKM} on the KP hierarchy, K…
Authors: Jipeng Cheng, Kelei Tian, Jingsong He
THE ADDITIONAL SYMMETRIES FOR THE BTL AND CTL HIERAR CHIES JIPENG CHENG 1 , 2 , KELEI TIAN 2 , JINGSONG HE 1 ∗ , 1 Dep ar tment of Mathematics, Ningb o University, Ningb o, Zhejiang 315211, P. R. China 2 Dep ar tment of Mathematics, USTC, Hefei,Anhui 230026 , P. R. C hina Abstra ct. In this pap er, w e construct the additional sy m met ries for th e T oda lattice (TL) hierarc hies of B typ e and C type ( t he BTL and CTL hierarchies), and show their algebraic structu res are w B ∞ × w B ∞ and w C ∞ × w C ∞ respectively . And also we discuss the generating functions of th e additional symmetries. Keywords : th e BTL and CTL hierarc hies, additional symm etries MSC2010 numbers : 17B8 0, 35Q51, 37K10, 37K30 1. Introduction The T o da lattice (T L ) hierarch y w as first introdu ced b y K .Ueno and K .T ak a saki in [1] to generalize the T o da lat tice equations [2]. Along the work of E. Date, M. Jimbo, M. Kashiw ara and T. Miwa [3] on the KP hierarch y , K.Ueno and K .T ak asaki in [1] d ev elop the theory for the TL h ierarc h y: its algebraic structure, the linearization, the bilinear iden tit y , τ fu nction and so on. Also the analog ues of the B and C t yp es for the TL h ierarc h y , i.e . the BTL and CTL hierarc hies, are considered in [1], which are corresp ondin g to infinite dimensional Lie algebras o( ∞ ) and sp( ∞ ) resp ectiv ely . In this p ap er, we will fo cus on the study of the additional symmetries for the BTL and CTL h ierarc hies. Symmetries h a v e b een playing vital role s in the study of the integ rable syste m. The additional symmetry [4 – 11] is one of the most important symmetries, whic h has t w o different expressions. On e of its expressions can b e traced bac k to the m aster s y m metry [12–18]. As an interesting generaliza tion of usual symm etries of partial d ifferen tial equati ons, master symmetries are introd uced in references [12] and further deve lop ed in references [13 – 18]. The master sy m metries are usually for the g iv en explicit soliton equations, whose r emark ab le feature is i ts dep en d ing explicitly on the space x and time t v ariables for 1 + 1-dimensional case. When considering the in tegrable hierarc hies, the master symmetries are usually called additional symmetries [4 ]. F or the KP hierarch y , the co rresp onding additional sy m metries are studied in references [5–9 ], which can b e used to form w ∞ algebra when acting on the linear problem, wh ile the ad d itional symmetry for the TL hierarc hy is in v estig ated in references [8, 10, 11], whic h f orms w ∞ × w ∞ algebra wh en acting on the linear p roblem. The other expression f or the additional sym metry is in th e form of Sato B¨ ac klund transformation [3] defined b y the v erte x op erator X ( λ, µ ) acting o n the τ function. These t w o differen t expressions c an b e ∗ Corresponding author. email:hejingsong@n bu.edu.cn; jshe@ustc.edu.cn. 1 link ed by so-called Adler-Shiota -v an Moerb ek e (ASvM) form ulas [6 – 10] for the contin uous int egrable systems (the KP hierarc h y) and also the discrete ones (the TL hierarc h y). By the ASvM f orm ulas, the associated algebra of additional symmetries ca n b e lifte d to its cen tral extension, the algebra of B¨ ac klund symmetries. The add itional symmetries are inv olved in so-c alled string equation a nd the generalized Virasoro constrain ts in matrix mo d els of the 2d qu an tum gra vit y (see [5, 7, 1 1] and references therein). There are several interesting results ab out the additional symmetries f or the BKP and CKP hi- erarc hies f r om the views of th e p ossible applications r elated to th e string equations. F or the BKP hierarc h y , the corresp ond ing add itional symmetry are constructed by T ak asaki [19] in the op erator form, and Virasoro constrain ts and the ASvM form ula ha v e b een studied b y Johan v an d e Leur [20, 21] using an algebraic formalism. R ecently , T u [22] ga v e an alternativ e pro of of th e ASvM form ula of th e BKP hierarc h y by u sing Dic k eys metho d [9]. And the corresp onding string equation was also con- structed by T u in [23]. As for the C KP hierarc h y , the additional symm etry a nd string equation w ere w ell constructed b y He in [24]. Insp ir ed b y these wo rks and the relation b et w een the TL hierarc h y and the KP hierarch y , we shall in this pap er establish the additional sym m etries for the BTL and CT L hierarc hies, and th en inv estig ate their corresp ond in g algebraic structures, an d study some in teresting prop er ties of them. This paper is organized in the follo wing wa y . In Secti on 2, we recall some basic kno wledge about the BTL and CTL hierarc hies. Then, w e construct the add itional symmetry for the BTL hierarch y and giv e their algebraic structure in Section 3. Next, in Section 4, th e ad d itional symmetry for the CTL hierarch y is also inv estigated and the corresp onding algebraic structure is sh o wn. A t last, w e dev ote Section 5 to some conclusions and discussions. 2. the BTL and CTL hierarchies In this section, we will review some basic facts ab ou t the BTL and CTL hierarchies in the st yle of Ad ler & v an Moerb ek e [8, 10]. One c an refer to [1] f or more details ab out the BTL and CTL hierarc hies. First, consid er the algebra D = { ( P 1 , P 2 ) ∈ gl(( ∞ )) × gl(( ∞ )) | ( P 1 ) ij = 0 f or j − i ≫ 0 , ( P 2 ) ij = 0 for i − j ≫ 0 } , whic h has the follo wing splitting: D = D + + D − , D + = { ( P, P ) ∈ D | ( P ) ij = 0 for | i − j | ≫ 0 } = { ( P 1 , P 2 ) ∈ D | P 1 = P 2 } , D − = { ( P 1 , P 2 ) ∈ D | ( P 1 ) ij = 0 for j ≥ i, ( P 2 ) ij = 0 for i > j } , 2 with ( P 1 , P 2 ) = ( P 1 , P 2 ) + + ( P 1 , P 2 ) − giv en b y ( P 1 , P 2 ) + = ( P 1 u + P 2 l , P 1 u + P 2 l ) , ( P 1 , P 2 ) − = ( P 1 l − P 2 l , P 2 u − P 1 u ) , where f or a matrix P , P u and P l denote th e upp er (including diagonal) and strictl y lo w er triangular parts of P , resp ectiv ely . F or ( P 1 , P 2 ) , ( Q 1 , Q 2 ) ∈ D , w e define ( P 1 , P 2 )( Q 1 , Q 2 ) = ( P 1 Q 1 , P 2 Q 2 ) , ( P 1 , P 2 ) − 1 = ( P − 1 1 , P − 1 2 ) . Then the BTL ( or CTL) hierarc h y is defin ed in the Lax forms as ∂ x 2 n +1 L = [( L 2 n +1 1 , 0) + , L ] and ∂ y 2 n +1 L = [(0 , L 2 n +1 2 ) + , L ] , n = 0 , 1 , 2 , · · · (1) where the Lax op erator L is giv en b y a pair of in fi nite matrices L = ( L 1 , L 2 ) = X −∞
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