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Symmetry algebras of quantum many-body systems with locality can beunderstood using commutant algebras, which are defined as algebras of operatorsthat commute with a given set of local operators. In this work, we show thatthese symmetry algebras can be expressed as frustration-free ground states of alocal superoperator, which we refer to as a "super-Hamiltonian". We demonstratethis for conventional symmetries such as $Z_2$, $U(1)$, and $SU(2)$, where thesymmetry algebras map to various kinds of ferromagnetic ground states, as wellas for unconventional ones that lead to weak ergodicity breaking phenomena ofHilbert space fragmentation and quantum many-body scars. In addition, we showthat this super-Hamiltonian is exactly the superoperator that governs theoperator relaxation in noisy symmetric Brownian circuits. This physicalinterpretation provides a novel interpretation for Mazur bounds forautocorrelation functions, and relates the low-energy excitations of thesuper-Hamiltonian to approximate symmetries that determine slowly relaxingmodes in symmetric systems. We find examples of gapped/gaplesssuper-Hamiltonians indicating the absence/presence of slow-modes, which happensin the presence of discrete/continuous symmetries. In the gapless cases, werecover slow-modes such as diffusion, tracer diffusion, and asymptotic scars inthe presence of $U(1)$ symmetry, Hilbert space fragmentation, and a tower ofquantum scars respectively. In all, this demonstrates the power of thecommutant algebra framework in obtaining a comprehensive understanding ofsymmetries and their dynamical consequences in systems with locality.

Symmetries as Ground States of Local Superoperators