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

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