  Deformation Theory is a natural generalization of Lie Theory, from Lie groups
and their linearization, Lie algebras, to differential graded Lie algebras and
their higher order deformations, quantum groups.
  The article focuses on two basic constructions of deformation theory: the
universal solution of Maurer-Cartan Equation (MCE), which plays the role of the
exponential of Lie Theory, and its inverse, the Kuranishi functor, as the
logarithm.
  The deformation functor is the gauge reduction of MCE, corresponding to a
Hodge decomposition associated to the strong deformation retract data.
  The above comparison with Lie Theory leads to a better understanding of
Deformation Theory and its applications, e.g. the relation between quantization
and Connes-Kreimer renormalization, quantum doubles and Birkhoff decomposition.
