# Holomorphic functions on the quantum polydisk and on the quantum ball

### Alexei Yu. Pirkovskii

National Research University Higher School of Economics, Moscow, Russia

## Abstract

We introduce and study noncommutative (or „quantized") versions of the algebras of holomorphic functions on the polydisk and on the ball in $C_{n}$. Specifically, for each $q∈C∖{0}$ we construct Fréchet algebras $O_{q}(D_{n})$ and $O_{q}(B_{n})$ such that for $q=1$ they are isomorphic to the algebras of holomorphic functions on the open polydisk $D_{n}$ and on the open ball $B_{n}$, respectively. In the case where $0<q<1$, we establish a relation between our holomorphic quantum ball algebra $O_{q}(B_{n})$ and L.L. Vaksman's algebra $C_{q}(Bˉ_{n})$ of continuous functions on the closed quantum ball. Finally, we show that $O_{q}(D_{n})$ and $O_{q}(B_{n})$ are not isomorphic provided that $∣q∣=1$ and $n≥2$. This result can be interpreted as a $q$-analog of Poincaré's theorem, which asserts that $D_{n}$ and $B_{n}$ are not biholomorphically equivalent unless $n=1$.

## Cite this article

Alexei Yu. Pirkovskii, Holomorphic functions on the quantum polydisk and on the quantum ball. J. Noncommut. Geom. 13 (2019), no. 3, pp. 857–886

DOI 10.4171/JNCG/340