  We consider the Lindblad equation for a collection of multilevel systems
coupled to independent environments. The equation is symmetric under the
exchange of the labels associated with each system and thus the open-system
dynamics takes place in the permutation-symmetric subspace of the operator
space. The dimension of this space grows polynomially with the number of
systems. We construct a basis of this space and a set of superoperators whose
action on this basis is easily specified. For a given number of levels, $M$,
these superoperators are written in terms of a bosonic realization of the
generators of the Lie algebra $\sln{M^2}$. In some cases, these results enable
finding an analytic solution of the master equation using known Lie-algebraic
methods. To demonstrate this, we obtain an analytic expression for the state
operator of a collection of three-level atoms coupled to independent radiation
baths. When analytic solutions are difficult to find, the basis and the
superoperators can be used to considerably reduce the computational resources
required for simulations.

Quantum Physics