The first part of this work focuses on the canonical group quantization approach applied
to non-commutative spaces, namely plane R2 and two-torus T2. Canonical
group quantization is a quantization approach that adopts the group structure that
respects the global symmetries of the phase space as a main ingredient. This is followed
by finding its unitary irreducible representations. The use of noncommutative
space is motivated by the idea of quantum substructure of space leading to nontrivial
modification of the quantization. Extending to noncommuting phase space includes
noncommuting momenta that arises naturally in magnetic background as in Landau
problem. The approach taken is to modify the symplectic structures corresponding
to the noncommutative plane, noncommutative phase space and noncommutative
torus and obtain their canonical groups. In all cases, the canonical group is found
to be central extensions of the Heisenberg group. Next to consider is to generalize
the approach to twisted phase spaces where it employs the technique of Drinfeld
twist on the Hopf algebra of the system. The result illustrates that a tool from the
deformation quantization can be used in canonical group quantization where the deformed
Heisenberg group H2
q is obtained and its representation stays consistent with
the discussion in the literature. In the second part, the two-parameter deformations of
quantum group for Heisenberg group and Euclidean group are studied. Both can be
achieved through the contraction procedure on SU(2)q;p quantum group. The study
also continues to develop (q; p)-extended Heisenberg quantum group from the previous
result. As conclusion, it is shown that the extensions of Heisenberg group arise
from quantizing noncommutative plane, noncommutative phase space, noncommutative
two-torus, and twisted phase space. The work on two-parameter deformation
of quantum group also further shows generalizations of the extension of Heisenberg
group
