  Much is understood about 1-dimensional spin chains in terms of entanglement
properties, physical phases, and integrability. However, the Lie algebraic
properties of the Hamiltonians describing these systems remain largely
unexplored. In this work, we provide a classification of all Lie algebras
generated by translation-invariant 2-local spin chain Hamiltonians, or
so-called dynamical Lie algebras. We consider chains with open and periodic
boundary conditions and find 17 unique dynamical Lie algebras. Our
classification covers some well-known models such as the transverse-field Ising
model and the Heisenberg chain, and we also find more exotic classes of
Hamiltonians that cannot be identified easily. In addition to the closed and
open spin chains, we consider systems with a fully connected topology, which
may be relevant for quantum machine learning approaches. We discuss the
practical implications of our work in the context of quantum control,
variational quantum computing, and the spin chain literature.
