  We study two-dimensional Hamiltonians in phase space with noncommutativity
both in coordinates and momenta. We consider the generator of rotations on the
noncommutative plane and the Lie algebra generated by Hermitian rotationally
invariant quadratic forms of noncommutative dynamical variables. We show that
two quantum phases are possible, characterized by the Lie algebras
$sl(2,\mathbb{R})$ or $su(2)$ according to the relation between the
noncommutativity parameters, with the rotation generator related with the
Casimir operator. From this algebraic perspective, we analyze the spectrum of
some simple models with nonrelativistic rotationally invariant Hamiltonians in
this noncommutative phase space, as the isotropic harmonic oscillator, the
Landau problem and the cylindrical well potential.
  PACS: 03.65.-w; 03.65.Fd
  MSC: 81R05; 20C35; 22E70
