  Non-commutative geometry indicates a deformation of the energy-momentum
dispersion relation $f(E)\equiv\frac{E}{pc}(\neq 1)$ for massless particles.
This distorted energy-momentum relation can affect the radiation dominated
phase of the universe at sufficiently high temperature. This prompted the idea
of non-commutative inflation by Alexander, Brandenberger and Magueijo (2003,
2005 and 2007). These authors studied a one-parameter family of
non-relativistic dispersion relation that leads to inflation: the $\alpha$
family of curves $f(E)=1+(\lambda E)^{\alpha}$. We show here how the
conceptually different structure of symmetries of non-commutative spaces can
lead, in a mathematically consistent way, to the fundamental equations of
non-commutative inflation driven by radiation. We describe how this structure
can be considered independently of (but including) the idea of non-commutative
spaces as a starting point of the general inflationary deformation of
$SL(2,\mathbb{C})$. We analyze the conditions on the dispersion relation that
leads to inflation as a set of inequalities which plays the same role as the
slow roll conditions on the potential of a scalar field. We study conditions
for a possible numerical approach to obtain a general one parameter family of
dispersion relations that lead to successful inflation.
