  In this work, we revisit several families of standard Hamiltonians that
appear in the literature and discuss their symmetries and conserved quantities
in the language of commutant algebras. In particular, we start with families of
Hamiltonians defined by parts that are local, and study the algebra of
operators that separately commute with each part. The families of models we
discuss include the spin-1/2 Heisenberg model and its deformations, several
types of spinless and spinful free-fermion models, and the Hubbard model. This
language enables a decomposition of the Hilbert space into dynamically
disconnected sectors that reduce to the conventional quantum number sectors for
regular symmetries. In addition, we find examples of non-standard conserved
quantities even in some simple cases, which demonstrates the need to enlarge
the usual definitions of symmetries and conserved quantities. In the case of
free-fermion models, this decomposition is related to the decompositions of
Hilbert space via irreducible representations of certain Lie groups proposed in
earlier works, while the algebra perspective applies more broadly, in
particular also to arbitrary interacting models. Further, the von Neumann
Double Commutant Theorem (DCT) enables a systematic construction of local
operators with a given symmetry or commutant algebra, potentially eliminating
the need for "brute-force" numerical searches carried out in the literature,
and we show examples of such applications of the DCT. This paper paves the way
for both systematic construction of families of models with exact scars and
characterization of such families in terms of non-standard symmetries, pursued
in a parallel paper.
