Fermionic one- and two-dimensional Toda lattice hierarchies and their bi-Hamiltonian structures

Abstract

By exhibiting the corresponding Lax pair representations we propose a wide class of integrable two-dimensional (2D) fermionic Toda lattice (TL) hierarchies which includes the 2D N = (2|2) and N = (0|2) supersymmetric TL hierarchies as particular cases. Performing their reduction to the one-dimensional case by imposing suitable constraints we derive the corresponding 1D fermionic TL hierarchies. We develop the generalized graded R-matrix formalism using the generalized graded bracket on the space of graded operators with an involution generalizing the graded commutator in superalgebras, which allows one to describe these hierarchies in the framework of the Hamiltonian formalism and construct their first two Hamiltonian structures. The first Hamiltonian structure is obtained for both bosonic and fermionic Lax operators while the second Hamiltonian structure is established for bosonic Lax operators only. We propose the graded modified Yang-Baxter equation in the operator form and demonstrate that for the class of graded antisymmetric R-matrices it is equivalent to the tensor form of the graded classical Yang-Baxter equation.

Bosonic Symmetries of the Extended Fermionic (2N, 2M)-Toda Hierarchy

“…With the above preparation, the N=(1|1) supersymmetric 2D Toda lattice hierarchy will be introduced as [26] in the next section.…”

“…Using the definitions of L n * ,L n * , we can easily obtain the following useful identities [26] L 2n * , L 2m…”

“…(3.18) Next, using (3.8-3.11), one can derive dynamical equations for the operators L n * ,L n * [26],…”

“…In this section, we construct the two-dimensional fermionic (K, S )-Toda lattice hierarchy in terms of the Lax-pair representation similarly as the fermionic (K, S )-Toda lattice hierarchy in [26]. Let us consider two difference operators…”

“…Here the logarithmic Lax operator log L has the even Grassmann parity. Under the reduction u 0 (x) = 1, one can derive the 1D (2, 2)-Toda lattice hierarchy in [26]. In this case, the representation (6.3) will be given by the Lax operator…”