  We consider a deformation of 3D lattice gauge theory in the canonical
picture, first classically, based on the Heisenberg double of
$\operatorname{SU}(2)$, then at the quantum level. We show that classical
spinors can be used to define a fundamental set of local observables. They are
invariant quantities which live on the vertices of the lattice and are labelled
by pairs of incident edges. Any function on the classical phase space, e.g.
Wilson loops, can be rewritten in terms of these observables. At the quantum
level, we show that spinors become spinor operators. The quantization of the
local observables then requires the use of the quantum $\mathcal{R}$-matrix
which we prove to be equivalent to a specific parallel transport around the
vertex. We provide the algebra of the local observables, as a Poisson algebra
classically, then as a $q$-deformation of $\mathfrak{so}^*(2n)$ at the quantum
level. This formalism can be relevant to any theory relying on lattice gauge
theory techniques such as topological models, loop quantum gravity or of course
lattice gauge theory itself.
