We introduce and study noncommutative (or „quantized") versions of the algebras of holomorphic functions on the polydisk and on the ball in . Specifically, for each we construct Fréchet algebras and such that for they are isomorphic to the algebras of holomorphic functions on the open polydisk and on the open ball , respectively. In the case where , we establish a relation between our holomorphic quantum ball algebra and L.L. Vaksman's algebra of continuous functions on the closed quantum ball. Finally, we show that and are not isomorphic provided that and . This result can be interpreted as a -analog of Poincaré's theorem, which asserts that and are not biholomorphically equivalent unless .
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. 857–886DOI 10.4171/JNCG/340