A Note on Fermionic Flows of the N=(1|1) Supersymmetric Toda Lattice Hierarchy

We extend the Sato equations of the N=(1|1) supersymmetric Toda lattice hierar-chy by two new infinite series of fermionic flows and demonstrate that the algebra of the flows of the extended hierarchy is the Borel subalgebra of the N=(2|2) loop superalgebra. In this note, we consider the integrable N=(1|1) supersymmetric generalization [1] of the two-dimensional bosonic Toda lattice hierarchy (2DTL hierarchy) [7]. It is given by an infinite system of evolution equations (flows) for an infinite set of bosonic and fermionic lattice fields evolving in two bosonic and two fermionic infinite “towers ” of times. A subsystem of the 2DTL hierarchy involves an N=(1|1) supersymmetric integrable generalization of the 2DTL equation, which is called the N=(1|1) 2DTL equation. Two new infinite series of fermionic flows of the N=(1|1) 2DTL hierarchy were con-structed in [3] in a heuristic way by solving symmetry equations corresponding to the N=(1|1) 2DTL equation derived in [5]. This hierarchy was shown to actually have a higher symmetry, namely an N = (2|2) supersymmetry [4, 2]. Together with the previ