The Additional Symmetries for the BTL and CTL Hierarchies

TL;DR

This paper investigates the additional symmetries of the BTL and CTL hierarchies, which are extensions of the Toda lattice hierarchy associated with orthogonal and symplectic Lie algebras.

Contribution

It provides a detailed analysis of the additional symmetries specific to the BTL and CTL hierarchies, expanding the understanding of their algebraic structures.

Findings

Identified new classes of additional symmetries for BTL and CTL hierarchies.

Established the algebraic relations governing these symmetries.

Connected the symmetries to underlying Lie algebra structures.

Abstract

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.