Generalized Toda Mechanics Associated with LoopAlgebra ℒ(Br)

Abstract

A class of generalization of Toda mechanics with long rangeinteractions is constructed in this paper. These systems areassociated with the loop algebras ℒ(Br) in the sensethat their Lax matrices can be realized in terms of the c = 0representations of the affine Lie algebras B(1)r. We adopt apair of ordered integers (m,n) to describe the Toda mechanicssystem when we present the equations of motion and the Hamiltonianstructure. We also extract the classical r matrix which satisfythe classical Yang–Baxter relation. Such generalizations willbecome systems with noncommutative variables in the quantum case.