  The forward BFKL equation is discretised in virtuality space and it is shown
that the diffusion into infrared and ultraviolet momenta can be understood in
terms of a semi-infinite matrix. The square truncation of this matrix can be
exponentiated leading to asymptotic eigenstates sharing many features with the
BFKL gluon Green's function in the limit of large matrix size. This truncation
is closely related to a representation of the XXX Heisenberg spin $= -
\frac{1}{2}$ chain with SL(2) invariance where the Hamiltonian acts on a
symmetric double copy of the harmonic oscillator. A simple modification of the
BFKL matrix suppressing the infrared modes generates evolution with energy
compatible with unitarity.
