Holomorphic functions on the quantum polydisk and on the quantum ball

Abstract

We introduce and study noncommutative (or „quantized") versions of the algebras of holomorphic functions on the polydisk and on the ball in ${\mathbb{C}}^n$. Specifically, for each $q\in \mathbb{C}\setminus \{0\}$ we construct Fréchet algebras ${\mathcal{O}}_q({\mathbb{D}}^n)$ and ${\mathcal{O}}_q({\mathbb{B}}^n)$ such that for $q=1$ they are isomorphic to the algebras of holomorphic functions on the open polydisk ${\mathbb{D}}^n$ and on the open ball ${\mathbb{B}}^n$, respectively. In the case where $0<q<1$, we establish a relation between our holomorphic quantum ball algebra ${\mathcal{O}}_q({\mathbb{B}}^n)$ and L.L. Vaksman's algebra $C_q({\bar{\mathbb{B}}}^n)$ of continuous functions on the closed quantum ball. Finally, we show that ${\mathcal{O}}_q({\mathbb{D}}^n)$ and ${\mathcal{O}}_q({\mathbb{B}}^n)$ are not isomorphic provided that $|q|=1$ and $n\ge 2$. This result can be interpreted as a $q$-analog of Poincaré's theorem, which asserts that ${\mathbb{D}}^n$ and ${\mathbb{B}}^n$ are not biholomorphically equivalent unless $n=1$.