Some examples of algebras of noncommutative holomorphic functions

Thursday 4-Dec-2025 

Abstract

Algebras of noncommutative holomorphic functions in a finite number of variables can be obtained as specific completions of finitely-generated associative algebras or, more generally, as quotients of algebras of free entire functions. We will consider examples such as quantum planes and Drinfeld-Jimbo algebras and provide a complete picture in the case when the modulus of the quantum parameter $q$ does not equal $1$. The case when $|q|=1$ is more complicated and requires further study. We will also consider completions of universal enveloping algebras of Lie algebras, particularly nilpotent ones, and examine their relationship with quantum groups.