  We discuss duality relations in correlated electronic ladder systems to
clarify mutual relations between various conventional and unconventional
phases. For the generalized two-leg Hubbard ladder, we find two exact duality
relations, and also one asymptotic relation which holds in the low-energy
regime. These duality relations show that unconventional (exotic) density-wave
orders such as staggered flux or circulating spin-current are directly mapped
to conventional density-wave orders, which establishes the appearance of
various exotic states with time-reversal and/or spin symmetry breaking. We also
study duality relations in the SO(5) symmetry that was proposed to unify
antiferromagnetism and d-wave superconductivity. We show that the same SO(5)
symmetry also unifies circulating spin current order and s-wave
superconductivity.
