In this article we study two-dimensional Dirac Hamiltonians with
non-commutativity both in coordinates and momenta from an algebraic
perspective. In order to do so, we consider the graded Lie algebra
$\mathfrak{sl}(2|1)$ generated by Hermitian bilinear forms in the
non-commutative dynamical variables and the Dirac matrices in $2+1$ dimensions.
By further defining a total angular momentum operator, we are able to express a
class of Dirac Hamiltonians completely in terms of these operators. In this
way, we analyze the energy spectrum of some simple models by constructing and
studying the representation spaces of the unitary irreducible representations
of the graded Lie algebra $\mathfrak{sl}(2|1)\oplus \mathfrak{so}(2)$. As
application of our results, we consider the Landau model and a fermion in a
finite cylindrical well.
