  We introduce algebraic approach for superoperators that might be useful tool
for investigation of quantum (bosonic) multi-mode systems and its dynamics. In
order to demonstrate potential of proposed method we consider multi-mode
Liouvillian superoperator that describes relaxation dynamics of a quantum
system (including thermalization and intermode coupling). Considered algebraic
structure of superoperators that form Liouvillian and their algebraic
properties allows us to diagonilize multi-mode Liouvillian to find its
spectrum. Also it allows to derive linear by mean number of thermal
(environmental) photons approximation for time-evolution superoperator that
keeps amount of considered dimensions in Fock space finite (assuming initial
amount of dimensions finite) that might be helpful regarding entanglement
dynamics problems. Conjugate Liouvillian is considered as well in order to
perform analysis in Heisenberg picture, it can be implemented for multi-time
correlation functions derivation.
