On the relationships between covariance and the decoherence-free subalgebra of a quantum Markov semigroup

n this paper, we investigate if the presence of symmetries in the evolution of an open quantum system gives information about the invariant subspaces. Unfortunately, the answer is, in general, negative, but under suitable conditions we can observe that the representation through which we describe the symmetry determines a privileged family of orthogonal projections strongly correlated with the structure of the decoherence-free sub-algebra and, as a consequence, with harmonic projections.