The lattice is described by a graph  with vertex set  and edge set .
The model has an associated Lie algebra . More generally, this Lie algebra can be taken to be any complex, finite-dimensional semi-simple Lie algebra . More generally still it can be taken to be an arbitrary Lie algebra.
Each vertex  has an associated representation of the Lie algebra , labelled . This is a quantum generalization
of statistical lattice models, where each vertex has an associated spin variable.
The Hilbert space  for the whole system, which could be called the configuration space, is the tensor product of the representation spaces at each vertex: 
A Hamiltonian is then an operator on the Hilbert space. In the theory of spin chains, there are possibly many Hamiltonians which mutually commute. This allows the operators to be simultaneously diagonalized. 
There is a notion of exact solvability for spin chains, often stated as determining the spectrum of the model. In precise terms, this means determining the simultaneous eigenvectors of the Hilbert space for the Hamiltonians of the system as well as the eigenvalues of each eigenvector with respect to each Hamiltonian.