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.Ueno and K.Takasaki in \cite{uenotaksasai} develop the theory for the TL hierarchy: its algebraic structure, the linearization, the bilinear identity, $\tau$ function and so on. Also the analogues of the B and C types for the TL hierarchy, i.e. the BTL and CTL hierarchies, are considered in \cite{uenotaksasai}, which are corresponding to infinite dimensional Lie algebras $\textmd{o}(\infty)$ and $\textmd{sp}(\infty)$ respectively. In this paper, we will focus on the study of the additional symmetries for the BTL and CTL hierarchies.Comment: 13 page

Additional symmetries of constrained CKP and BKP hierarchies

The additional symmetries of the constrained CKP (cCKP) and BKP (cBKP) hierarchies are given by their actions on the Lax operators, and their actions on the eigenfunction and adjoint eigenfunction $\{\Phi_i,\Psi_i \}$ are presented explicitly. Furthermore, we show that acting on the space of the wave operator, $\partial_k^*$ forms new centerless $W^{cC}_{1+\infty}$ and $W^{cB}_{1+\infty}$-subalgebra of centerless $W_{1+\infty}$ respectively. In order to define above symmetry flows $\partial_k^*$ of the cCKP and cBKP hierarchies, two vital operators $Y_k$ are introduced to revise the additional symmetry flows of the CKP and BKP hierarchies.Comment: 14 pages, accepted by SCIENCE CHINA Mathematics(2010