Algebras of noncommutative holomorphic functions: examples

Algebras of noncommutative holomorphic functions of finitely many variables can be obtained as specific completions of finitely generated associative algebras or, more generally, as quotients of the algebras of free entire functions. We consider examples such as quantum planes and Drinfeld-Jimbo algebras, and present a complete picture for the case where the modulus of the quantum parameter $q$ is not equal to $1$. The case where $|q|=1$ is more complex and requires further study. We also discuss completions of universal enveloping algebras of Lie algebras, in particular, of nilpotent ones.