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