Recent advances in quantum simulator platforms allow for realizing many-body systems coupled to an environment, opening the possibility for novel mixed-state phases of matter and dynamics. In this talk, I will present two results exploring the role of symmetries in this new paradigm. First, I will discuss a partial classification of 2D mixed-state topological order, which may be characterized through higher-form symmetries. Topologically ordered phases of matter support fractionalized anyon excitations. We show that subjecting these states to decoherence has the effect of "gauging out" certain anyons and leads to mixed-state phases which, as pure states, can only be realized on the surface of 3D topologically ordered states. Second, I will discuss how conventional symmetries can lead to non-trivial constraints, in the spirit of Lieb-Schultz-Mattis theorems, on the dynamics of open many-body systems governed by Lindbladian evolution, which may have implications for dissipative state preparation.
