  We propose a quantization of linear, volume preserving, maps on the discrete
and finite 3-torus T_N^3 represented by elements of the group SL(3,Z_N). These
flows can be considered as special motions of the Nambu dynamics (linear Nambu
flows) in the three dimensional toroidal phase space and are characterized by
invariant vectors, a, of T_N^3. We quantize all such flows which are
necessarily restricted on a planar two-dimensional phase space, embedded in the
3-torus, transverse to the vector a . The corresponding maps belong to the
little group of the vector a in SL(3,Z_N) which is an SL(2,Z_N) subgroup. The
associated linear Nambu maps are generated by a pair of linear and quadratic
Hamiltonians (Clebsch-Monge potentials of the flow) and the corresponding
quantum maps, realize the metaplectic representation of SL(3,Z_N) on the
discrete group of three dimensional magnetic translations i.e. the
non-commutative 3-torus with deformation parameter the N-th root of unity.
  Other potential applications of our construction are related to the
quantization of deterministic chaos in turbulent maps as well as to quantum
tomography of three dimensional objects.
