  The magnetic translation algebra plays an important role in the quantum Hall
effect. Murthy and Shankar, arXiv:1207.2133, have shown how to realize this
algebra using fermionic bilinears defined on a two-dimensional square lattice.
We show that, in any dimension $d$, it is always possible to close the magnetic
translation algebra using fermionic bilinears, whether in the continuum or on
the lattice. We also show that these generators are complete in even, but not
odd, dimensions, in the sense that any fermionic Hamiltonian in even dimensions
that conserves particle number can be represented in terms of the generators of
this algebra, whether or not time-reversal symmetry is broken. As an example,
we reproduce the $f$-sum rule of interacting electrons at vanishing magnetic
field using this representation. We also show that interactions can
significantly change the bare bandwidth of lattice Hamiltonians when
represented in terms of the generators of the magnetic translation algebra.
