  We develop a topological classification of non-Hermitian effective
Hamiltonians that depend on momentum and frequency. Such effective Hamiltonians
are in one-to-one correspondence to single-particle Green's functions of
systems that satisfy translational invariance in space and time but may be
interacting or open. We employ K-theory, which for the special case of
noninteracting systems leads to the well-known tenfold-way topological
classification of insulators and fully gapped superconductors. Relevant
theorems for K-groups are reformulated and proven in the more transparent
language of Hamiltonians instead of vector bundles. We obtain 54 symmetry
classes for frequency-dependent non-Hermitian Hamiltonians satisfying
anti-unitary symmetries. Employing dimensional reduction, the group structure
for all these classes is calculated. This classification leads to a group
structure with one component from the momentum dependence, which corresponds to
the non-Hermitian generalization of topological insulators and superconductors,
and two additional parts resulting from the frequency dependence. These parts
describe winding of the effective Hamiltonian in the frequency direction and in
combined momentum-frequency space.
