Quantum polydisk, quantum ball, and a q-analog of Poincaré's theorem

Abstract

The classical Poincaré theorem (1907) asserts that the polydisk ${\mathbb{D}}^n$ and the ball ${\mathbb{B}}^n$ in ${\mathbb{C}}^n$ are not biholomorphically equivalent for $n\ge 2$. Equivalently, this means that the Fréchet algebras $\mathcal{O}({\mathbb{D}}^n)$ and $\mathcal{O}({\mathbb{B}}^n)$ of holomorphic functions are not topologically isomorphic. Our goal is to prove a noncommutative version of the above result. Given $q\in \mathbb{C}\setminus \{0\}$, we define two noncommutative power series algebras ${\mathcal{O}}_q({\mathbb{D}}^n)$ and ${\mathcal{O}}_q({\mathbb{B}}^n)$, which can be viewed as $q$-analogs of $\mathcal{O}({\mathbb{D}}^n)$ and $\mathcal{O}({\mathbb{B}}^n)$, respectively. Both ${\mathcal{O}}_q({\mathbb{D}}^n)$ and ${\mathcal{O}}_q({\mathbb{B}}^n)$ are the completions of the algebraic quantum affine space ${\mathcal{O}}_q^{\mathrm{reg}}({\mathbb{C}}^n)$ w.r.t. certain families of seminorms. In the case where $0<q<1$, the algebra ${\mathcal{O}}_q({\mathbb{B}}^n)$ admits an equivalent definition related to L. L. Vaksman's algebra of continuous functions on the closed quantum ball. We show that both ${\mathcal{O}}_q({\mathbb{D}}^n)$ and ${\mathcal{O}}_q({\mathbb{B}}^n)$ can be interpreted as Fréchet algebra deformations (in a suitable sense) of $\mathcal{O}({\mathbb{D}}^n)$ and $\mathcal{O}({\mathbb{B}}^n)$, respectively. Our main result is that ${\mathcal{O}}_q({\mathbb{D}}^n)$ and ${\mathcal{O}}_q({\mathbb{B}}^n)$ are not isomorphic if $n\ge 2$ and $|q|=1$, but are isomorphic if $|q|\ne 1$.

16 pages

References in corpus (3)

• The isomorphism problem for some universal operator algebras
• Noncommutative complex geometry of the quantum projective space
• Quantum Bundle Description of the Quantum Projective Spaces