Super toda lattices

Abstract

The Lax formalism, as described by Oevel et al. and in an earlier and more fundamental form by Semenov, Kostant, Symes, and Adler, can easily be generalized to the case where anticommuting variables are involved, the so‐called supercase. In this article this super‐Lax formalism is applied to the well‐known associative superalgebra G=Mat(m,n,Λ). Subspaces of G to which the super‐Poisson structures can be restricted arise in a natural way. Taking L in one of these subspaces formally leads to superextensions of the hierarchy of nonrelativistic Toda lattices. In the simplest case, where only nearest‐neighbor interaction is involved, the equations are explicitly solved. Furthermore, the relevant two super‐Hamiltonian structures are explicitly calculated. Finally a superextension of the relativistic Toda lattice with a super‐Hamiltonian structure is described herein.