Every Lie group $G$ carries a distinguished algebra of particularly well-behaved real-analytic mappings: The entire functions $\mathcal{ E}(G)$. They were introduced for the purposes of strict deformation quantization. This paper establishes a one-to-one correspondence between entire functions and holomorphic mappings $\mathcal{ H}(G_\mathbb{ C})$ on the universal complexification $G_\mathbb{ C}$ of $G$ as Fr\'{ e}chet algebras. Methodically, this is achieved by porting aspects of classical complex analysis into a left-invariant guise and by studying the geometry of $G_\mathbb{ C}$. As a byproduct, we obtain a strict deformation quantization of the holomorphic cotangent bundle of any universal complexification.
Categories: math.CV